I like your budget DIY approach - very cool project. There are a couple of new 22s that come to mind - the key is that they break doen and each have a soft carry case that could be a mini bugout bag as well:
-Ruger 10/22 Takedown 22LR 18.5" - 10 +1, 4.6lb. with soft carry case
-MARLIN PAPOOSE MODEL 70PSS, SEMI AUTO RIFLE, 22 LR, TAKE-DOWN W/CASE, 7+1 with soft carry case
Both seem to be in short supply - probably due to a lot of folks thinking like yourself. The new 22s will run 300 or more depending BUT they are semi autos.
Marlin review: Good for backpacking...Sturdy, pack-able, light, very accurate. And that removable barrel makes it a breeze to clean. I keep getting mine seriously wet but that great Marlin stainless keeps things the way they ought to be.
Ruger: First of all the bag it comes in, is top notch quality. It feels well made and holds the rifle, a scope, 3 bx-25 mags, and even some spare parts and ammo without any trouble and the sling feels sturdy(however untested in long term use). The 10/22 takedown really is as simple as it looks to takedown and re-assemble, and has seemed to hold its zero every time. I installed the scope rail with easy and everything else is just simple 10/22 reliability. You cannot go wrong with this purchase!