The strain gauge output isn't negative, exactly. Its essentially a wheatstone bridge, and the measurement is called a ratiometric measurement. Essentially, what changes is the resistance of the actual sensing element. The wheatstone bridge allows you to measure small changes in resistance.
The bridge consists of 4 equal resistances when theres no strain. For a quarter bridge, one of the legs, say the top right one when the bridge looks like a diamond is the strain sensing element. The other resistances are chosen to be equal to the strain sensor resistance. A half and full bridge have 2 opposite and all 4 legs with strain sensors, and can be more accurate because resistor tolerance is lower than strain sensor error. I'll explain the answer for a quarter bridge.
Now, when you excite the bridge by applying 10V across the top and bottom points, you produce half the voltage at the two central points of the bridge. Note that i haven't mentioned ground anywhere. If you excite using +/-5, the centre nodes end up at zero. If you excite using 10V and zero, the centre nodes end up at 5v.
Now, when strain is applied to the one sensing element, the bridge becomes unbalanced, creating a small difference between the voltages at the center (sensing) nodes. This difference is what you can measure easily.
Use a single supply instrumentation amplifier rated to run at 10V single supply. Instrumentation amplifiers are essentially amplifiers which amplify the difference between two signals. They also have a ref input, which allows you to add an arbitrary external voltage to the output.
Lets say \$V_1\$ and \$V_2\$ are the bridge sensing node voltages. \$G\$ is the gain of the instrumentation amplifier, and Vref is the ref input of the instrumentation amplifier.
The output is then \$G \cdot (V_1 - V_2) + V_{ref}\$.
If you send in \$V_{ref}\$ of 5V (half the excitation voltage) , using a simple voltage divider and an op amp buffer (also single supply), and ensure \$G\$ is small enough for \$G \cdot (V_1 - V_2)\$ is less than 5 volts, you end up with a single ended positive output in the range of 0 to 10V, which you can read using a regular single ended ADC.
There are other minor implementation details you should take care of. For instance, the excitation voltage should be cleaner than most usual regulators can provide. Something based on a band gap reference and buffered using a transistor is a better bet, if possible (your supply would have to be greater than the excitation voltage). You would also have to ensure the full scale range of your ADC is sufficient, or set your excitation voltage accordingly.
This is probably a considerably better solution than trying to make a symmetric supply, which is fraught with unnecessary complexity and high potential for unwanted noise to creep in. If you insist, though, the solution thats most straightforward would be to use a virtual ground instead, which you can make by a potential divider halving the supply voltage and an opamp buffer (voltage follower), probably augmented by a transistor buffer stage to supply the necessary current.
Coin cells will not work well with your needs. While it varies by brand and size, most coin cells have low capacity, and an internal resistance that limits current output. A CR2032 has a nominal 250ma capacity (and that varies based on current load btw), and an internal resistance (Equivalent Series Resistance or ESR) of 18 to 30ohms, again based on current load). Depending on the speed of the current draw, you are going to get voltage drops, and quickly drain the coin cell. If you can get that much current out of it at a given time. 70mA is pushing it even for high quality CR2032, and 220mA is improbable, frankly, impossible. That is why your CR2032 boosted to 3.3v isn't working. The current draw of the regulator is causing the cell's voltage to drop.
That is, without having multiple in parallel (or using high capacity caps). That's the second part you asked. Batteries in series are a sum of the voltages, with the same amount of current. Batteries in parallel are a sum of the current capacities, with the same voltage.
As for your options, one or two AAA batteries with a boost converter would suit you better. It gives you relatively high capacity while still keeping space to a minimal. The TPS61200 you mention would work, if you can deal with a tiny (3x3mm leadless package. The TPS6107x family would do the same, same minimal components, in a sot6 package. You only need 1 battery for either of these. An AA gives 2400mAh nominal, an AAA gives 1200mAh nominal.
Best Answer
I would use a boost regulator form 3v to 5v, and then a voltage inverter like the Analog Devices ADM8829 or the Maxim MAX1697, about $2.30 in single quantities, to get the -5v.
Both take an input up to 5.5v, and produce a negative voltage of the same magnitude, with an output of either 25 mA or 60 mA which should be enough for op-amps. They only need a minimum of external components (two caps). Quiescent current is 600 uA for the AD part, and a few hundred µA for the Maxim part. Very simple, you don't have to deal with virtual grounds or the like.