For safety during transportation and storage, from the factory to the warehouse to the store, one would like the battery to be mostly discharged so that it's holding less stored energy in case of some mishap. If a forklift spears a big box of product, you'd like a minimal amount of energy released from the dead short which may be caused by crushing or puncturing cells. Probably there will be some protection built into both the cells and the product to try to prevent fire, however it's still safer to have less energy to release. Airlines in particular may have rules or laws which describe how batteries should be transported.
For purposes of maximizing lithium battery lifetime, it's better to keep them charged between 20-80% of full capacity most of the time. It's very damaging to over-discharge a battery and may render the cells useless. While any product shipping with a lithium battery should have circuitry to prevent over-discharge, it's possible the load on the battery will never quite be reduced to zero even when the product is "off" or in full shutdown, as there could still be a low-power microcontroller watching for a press of the power-on button, some tiny current powering the battery protection circuitry, etc. While this circuitry might run "indefinitely" from a full battery, if the battery is run down to empty and then the product is left in a drawer for a year it's possible the battery could get over-discharged regardless of the circuitry trying to prevent it.
One possible scenario could be that the factory charges the battery, the device goes into inventory for over a year (due to the product not selling well), the battery loses most of its charge by self-discharge and a small amount of current powering the "off"-mode circuitry e.g. microcontroller. Then if the device is turned on with this nearly-dead battery in it, the large pulse of discharge current causes the battery voltage to dip to where the undervoltage protection triggers, which may or may not turn off the current before the battery has been run below its minimum voltage for some small amount of time, like milliseconds. It probably won't destroy the cell but isn't a good idea. Especially if the user sees the product doesn't turn on and keeps hitting the power button a dozen times before deciding it must need to be charged up.
To minimize tech support phone calls, it's better to tell the user to fully charge their battery before trying to use the product, so you don't get calls about the product "not working" when the actual problem is that the battery is simply empty. It's one less variable in debugging whether the product is faulty. A product may have features that take more power than the baseline mode, and might turn on and operate properly, but then appear to fail when a higher-power feature is turned on (when in fact it's shutting down or resetting). Turning on the backlight in an otherwise simple device, or wireless transmission, loud audio, fan or motor, anything that's a large percentage of power consumption. If the battery is fully charged, it will have enough energy to power all modes, so this problem is avoided while talking to tech support.
That said, there's a lot of mythology out there regarding how batteries should be handled. The customer care person is probably not an engineer, and may be going by some mix of training, passed-around heuristic, and made-up baloney that's designed to mitigate unhappy boss or unpleasant customer rather than detect "the problem".
What you are trying to achieve does not seem overly hard but it needs to be better defined than has been done here so far because the situation that you describe sounds to be different enough than the usual charge & run situation that you need to make specific provisions.
You need to identify desired pass through Imax and I_pass_through_average for a given Ah capacity. The most demanding case will be that with the largest pass_through_Ah ratio.
Usually with simultaneous 'charge and run' situations I load is less or much less than Ichg_max. In such cases it is often OK to address the battery needs and accept that the load may somewhat prolong charging times. Clearly I_load_average must be less than I_charge_averge or the battery would never charge, but you may get situations where I_load is > I_charge for significant periods, or is a substantial fraction of I_Charge_terminate, and this will affect charging results.
Usually Imax in constant current mode Icc is the same in mA as the capacity in mAh. So a 1000 mAh cell is charge at 1000 mA max, a 5Ah cell at 5 A max etc.
If you just increase charging capacity without regard to interaction with charge termination you may 'have problems'. This is because LiIon/LiPo charging typically terminates when I charge in CV mode drops to some preselected percentage of ICC = Imax and I_pass_through is seen as part of Iterminate if special steps are not taken to prevent this.
Taking the extremes of your spec ie 3A charge and 5Ah cell that is 0.6C charge (still under the usual 1C max) the usual charging terminate current will usually be in the 0.5C to 0.1C range with 0.25C being typical. So here with a 5 Ah cell Iterminate will be 2.5 A / 1.25A / 0.5A max/typ/min. If I_pass_through is say 1A mean it will swamp a 0.5A terminate current and interact significantly with a 1.25A or even 2.5A terminate current. In a dumb charger, if the C/10 rate was used for terminate the charging would almost certainly never cease, at 1.25A terminate (C/4) it may cease on load dips and at 2.5A terminate charging may be prolonged.
A LiPo cell charged at CC of C/1 will reach about 75%-85% of full capacity when CV level of 4.2V is reached. If charging is at less than C/1 (as will be the case if you charge a 10 Ah cell at 5A or C/2) then an even larger % of the full capacity has been transferred when you reach 4.2V. If you are happy to accept say 90% of full capacity then the easy choice is to hard terminate charge when 4.2V is reached and have no CV final stage. This removes the need to make end of charge determinations based on combined pass through and CV currents. Some charger IC's are available with no CV stage.
Also, as now you only need a CC stage and a voltage cutoff at 4.2V then a current limited
power supply and a comparator will fill your need. Adjustment of the current limit on an off the shelf supply may suffice in a small volume research situation.
Best Answer
Over-discharging any rechargeable battery chemistry can ruin the cell(s). Depending on how far below minimum the voltage is allowed to go, and for how long, the effect could range from somewhat reduced capacity & cycle life, to an unusable cell.
How likely is the user to remember to recharge the battery before it's empty?
If the product is in deep sleep most of the time, and wakes rarely to do something very low-power, a single full charge of the battery may last for months, even years. Likely to be ignored until it stops working due to low battery.
There are several approaches.
1) Under-voltage protection circuit that effectively disconnects the battery from the circuit board (load), and keeps it disconnected until it has seen the battery go above threshold or receive a minimum amount of charge. An additional chip at extra hardware cost, but maybe less software work.
2) Software in MCU could monitor battery voltage and lock out full wakeup (running main code or turning on circuit board features) after low battery voltage, until it has sensed a power adapter input applied, or battery voltage above threshold. If MCU doesn't have ADC, can use low-cost voltage comparator into an input pin.
3) Device could warn user of moderately low battery (10-20% capacity remaining) by an intermittent beep, blinking light, network communication message. If user ignores all warnings, they may need to replace the battery.
4) Add small solar panel or other energy capture, so the device can run perpetually under nominal conditions. Manual battery recharge or replacement may still be necessary if a particular device is placed where there's not enough light.
Alternately,
5) Use a lithium primary battery large enough to last a long time. The product becomes 'disposable' and is replaced (recycled) entirely when the battery runs out. If the battery gets over-discharged, it doesn't matter, so long as it doesn't leak or cause other physical problems. For example, smoke detectors now come with lithium primary large enough to last 10 years, instead of expecting the user to swap a 9V every year or two.