Your concern is well-founded. If the battery is connected directly to the system load, there is no way for the charger to know how much current is going to the battery and how much is being used by the system.
For this reason, many chargers isolate VBAT from the system power rail (VSYS). In general, I would say that it is acceptable to put the system load in parallel with the battery when the system current is much smaller than the charge current, OR if the system current is very consistent. In that case, you can just crank up the charge current setting on the charger to account for the system current. But if the system current varies over a wide range, or is not consistent from unit-to-unit, I would look for a different charger IC.
As a side note, it appears that the MCP73831 does not have a charge timer. In other words, the only way for it to terminate charging is when the acceptance of the battery becomes sufficiently low. Normally, state of the art chargers would also have a timer, so that if the battery acceptance never becomes sufficiently low, charge will terminate anyway after allowing a generous interval of time. This lack is even more worrisome since you cannot separate the battery from the system load. The system load will make the battery acceptance appear higher than it really is, and the charger may never terminate, but just float at 4.2 V forever.
I recommend you ask Microchip about this, in case I am mistaken. But if it really is the case, then I have to suggest you look for another charger, because it is not safe to float lithium ion or polymer cells indefinitely at 4.2V. There needs to be a timer so that charge always terminates.
TI uses the term "power path" to describe chargers which keep the battery separate from the system. They are not the only ones with this feature, but I think searching for power path will help you to find chargers that do this, from TI and other vendors.
Yes, it is feasible. You just need to expose the balance plug and main positive and negative terminals (using a socket that cannot be accidentally shorted out or have something inappropriate plugged into it).
One possible problem is that the charger detects a full battery by monitoring the charge current and shutting off when it reaches 1/10th of the programmed current. At the standard charge rate of 4A the charger expects to shut off at ~400mA. However if your load is drawing 500mA then the charger may not properly detect the end of charge. While this won't damage the battery, the charger could run for longer than it needs to. Luckily your charger has a safety timer that can be set to 1-120 minutes (90 minutes should be plenty long enough).
You can charge through the balance connector by paralleling the positive and negative balance wires (red and black) with the main charger outputs. However the balance plug and wiring is only good for low current (<2A) so the charging time will be longer.
Note that during charging the battery voltage will reach (or even slightly exceed) 8.4V for quite a long period of time. Make sure that your load can handle this.
Best Answer
Do I need to limit the current progressively as the charge progresses? Yes. My understanding is that the battery will 'accept' lower and lower current as it nears the end of charge. You will need to be able to recognize that and then choose a value at which the battery is full. For example when charging current becomes 10% of original charging current then battery is full. For a faster charge, some people might choose a larger percentage as a cutoff.
Do I cutoff charging when voltage reaches 25.2 volts? As you can tell from the answer to number 1, no. To charge the lithium chemistry you start with a constant CURRENT until the battery reaches final voltage (25.2) then you charge at constant VOLTAGE allowing the current to slow down as the battery fills up, terminating charge as described in question 1.
Any IC to build charger around? Yes, lots. You do need to know how to build a charger around the IC but look here for an example. http://www.ti.com/product/bq24616. I am sending this via my phone so please excuse me if this link doesn't work. Just go to the TI.com website and search under power management and you will find around 200 to choose from.
Finally, the BMS that you are going to use could technically control the termination of the charging of the battery and it will in fact provide the all important balance charging that your battery will require, but most people recommend using it only as a backup to a proper battery charger and not to rely on the BMS as a battery charger.
Good luck.