I have a 240V subpanel on one side of my house which is fed from a main panel on the opposite side of the house. It is on a 100A breaker. This subpanel is full, with 2 50A breakers and a 30A breaker to an oven and 2 A/C condensers. I would like to somehow add a 20A breaker to this circuit to power a small 240V mini split A/C unit for my office, if possible.
The questions: Can the existing 100A circuit feeding this subpanel handle another 20A circuit? If so, would I need to add an additional subpanel fed by this one? Not to sound stupid, but is there a way to safely add it to the existing subpanel with some kind of tandem breaker or something? Maybe this: HOMT220230CP?
Edit:
Spec plates (what I could find anyway…)
Oven: 
A/C 1: 
I think that model # is N4H348AKF200 – I wasn't able to get a good angle as it's too close to the adjacent A/C unit.
A/C 2: 
The mini-split hasn't arrived yet but here is a link to the listing which has some technical specs: SENL-12CD/220/X
Best Answer
That 2-2-2-4 is just a trifle too small for your plans...
Were this an actual 100A subpanel, fed by 1AWG hot and neutral wires, you'd be able to do this without much issue. However, since you're dealing with 2AWG Al here, your feeder is only good for 90A due to the fact the 83% service/feeder rule for entire dwelling units doesn't apply to feeders that serve parts of dwelling units unless your service is the same size as the feeder, which I seriously doubt's the case in your situation. This hasn't become an issue yet since the feeder's NEC calculated load is under 90A as it stands at the moment; however, adding the minisplit is the proverbial straw that breaks the camel's back, pushing you to 93.5A (and a bit of change).
As a result (the load calcuation comes out to 1.25 * the wattage of the largest air conditioner + the wattage of the remaining air conditioners as computed by their rated amperage times 230V + the watt rating of the oven), you'll need to upgrade the SE cable from a 2-2-2-4 to the correct cable, namely a 1-1-1-3, before you proceed further with this.
...but the good news is that the breaker you need is a thing
Once that's out of the way, we can then fit the quadplex ("quad tandem" in Square-D lingo) Homeline breaker you will need, namely a HOMT220230, in place of the existing HOM230 serving the smaller of the two air conditioners. From there, it's just a matter of running the flex whip from the new outdoor unit to the breaker box and landing the wiring appropriately.