TLDR: Can't do that.
#1 Don't throw good money after bad! Old window A/Cs that are 240V are too inefficient to continue using. It's not about the environment, the electric bill will be so high it will actually be cheaper, all told, to buy a new one! Really. They're not serviceable either: you can't get the old freon, as it's no longer made. So an old unit is likely missing some gas and thus even more inefficient. Please - take it to where your city says to recycle air conditioners. A professional needs to vampire the freon out of it to keep it from damaging the atmosphere - and it will be re-refined into new freon (this is now the only source). That's Federal law by the way.
How is neutral going to work?
You have Plug-On Neutral breakers, so the neutral pigtail isn't in your face making you think about it. However the breakers still have neutral terminals, and you need to wire them correctly.
Most AFCI breakers, including these, have a "weak GFCI" section which is there to detect hot-ground and neutral-ground parallel arc faults (which also happen to be ground faults). Europe uses RCDs (weak GFCIs) for the same purpose. As such, you must follow "GFCI rules" for wiring.
Each AFCI will see normal machine current on the hot, but no similar current will not return on the neutral. So it will immediately trip. You cannot gang two GFCIs (or GFCI-using AFCIs) in this way.
GE has a GFCI that doesn't use neutral (except to power its own internals). It's designed for exactly this use case, but you can't use it in this panel. (you could get a 2-space panel and have the world's most expensive AFCI breaker).
Siemens also has (believe it or not) tandem AFCIs that don't use neutral, so you could get two tandems and handle-tie the "inners" in the normal way. (cost more than a new air conditioner LOL, talk about good money after bad!!!)
Siemens seems to have designed this breaker to reject any possible use of handle-ties. Makes sense, as the use-case to tie this type is extremely rare.
Other than that, Mrs. Lincoln, you can handle-tie this
- You can handle-tie loads which are 240V-only and do not receive a neutral.
- You can handle-tie "Multi-Wire Branch Circuits" (MWBC, shared neutral) as long as all outlets are 120V-only.
You can't get common trip with a handle-tie. (that comes from an internal mechanism). Breakers "trip free" (if handle is locked "on") so handle ties wouldn't do that anyway (as Longneck attests in a comment). Handle ties assure maintainers turn off the whole circuit so they don't get bit.
Where you need common trip is when a circuit or device has 120V loads, but it also has 240V loads. (A dryer, or MWBC including 240V outlets, which is allowed). You need to trip together because if one phase trips the breaker, the 240V load becomes a resistor connecting the dead leg to the live one, energizing it again! (just like what happens when a house loses a hot leg from the utility, but the dead 120V side works when you turn the oven on.)
