I'm looking for advice on the best way to fill a gap (about 1/2 inch wide) between the drywall and the window frame before re-sealing it with caulk. I've read that you can use a backer rod, or fill it with joint compound, or use insulating foam (specifically designed for windows) like the one found here (https://www.homedepot.com/p/GREAT-STUFF-12-oz-Window-and-Door-Insulating-Spray-Foam-Sealant-230612/202893747).
Any advice would be greatly appreciated.
You can use a non-expanding foam made for windows and doors or you could use some backer rod and an elastomeric caulking like DAP or Big Stretch.
A non-expanding foam will do just as it says, fill the gap without expanding and pinching or binding the window frame which would cause issues opening the window.
Elastomeric caulkings are designed to stretch without cracking and splitting. When working as a painter we used a lot of DAP products and the Big Stretch which is made for bigger gaps like yours.
Personally, I would use the foam first, trim it back once dry and then use the Big Stretch and finally some paint (if desired).
I'd use the backer rod.
The normal thing is to seal this gap for air tightness and then trim to hide the gap.
Spray foam is expensive for a small job and eventually cracks so it isn't the best product for an air seal.
If you are using trim then you don't need to worry about how the caulk looks when applied over the backer rod and you get the two sided hour glass adhesion that is the best practice to prevent three sided bonding and cracking. I used various thicknesses of backer rod, energy seal and then a passive house tape ( tescon vana ) before applying my trim.
