Will my electrical sub panel handle a HVAC load
You'll need to swap that BR260 in your garage panel for a BR290 to do this

Right now, given that the house panel serves 1000 ft2 at 3VA/ft2, as well as three small appliance branch circuits for the kitchen at 1500VA apiece, and an 8kVA allowance for your range, we plug those numbers into the standard Article 220 load calculation procedure and get about 52-53A of load on this panel out of it.  Since your AC pulls 15A of nameplate current, we tack that on and get about 66-67A of load in the new configuration, too much for the existing 60A breaker.

However, the fact that your SER cable is 2AWG is the saving grace here.  A 2AWG aluminum SER cable running between two pieces of distribution equipment with 75&deg;C rated lugs is good for 90A; I suspect it was breakered at 60A simply because the electrician didn't have any breakers between 60A and 100A (or even simply above 60A) on their truck when they did the electrical update.  So, we can free up some wiggle room here by swapping the BR260 in your garage panel for a BR290.

From there, you can fit a BR220 in the house panel (or a GFTCB220 if you want to be ready for the 2020 NEC's expanded GFCI requirements), and run 14/2 or 12/2 (your choice since the minimum circuit ampacity on your AC is 15A, and air conditioners use the Article 440 wire and breaker sizing rules, which say "follow the nameplate" for everything we're concerned with here) to the air conditioner disconnect box.  You'll need to transition to individual THHNs of the same gauge in liquidtight or a prefabricated flex whip for the disconnect box to air conditioner connection, by the way.