Electronic – Best way to power a 3.3V circuit from 3.3V-to-5.0V source

There are number of companies that offer drop in switching regulators that might meet your needs. I'd look at some of the electronics distributors out there with good parametric search capabilies like Digikey, Newark, or Mouser that might have something already designed that will work for you.

Although it's a little more expensive than your $3 cost target. Look at this drop-in switching regulator.

It handles the output current and input voltage range you've specified. It's $5.10 in quantities of 100. Take into consideration the time saving cost benefit of not having to design one yourself.

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.