Electronic – Sensible way to power an OLED display from AAA Cells

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.

Solderless breadboards are useless for prototyping circuits over a few hundred kHz. For high frequencies, you need to construct your prototype circuits "deadbug" style over a copper clad ground plane, like so: