The purse makes more sense, they do have backpack style carriers like they have for children that shouldn’t throw you off balance like just a purse does.
The strollers a fun idea but it is bulky and can’t be stashed or put aside easily if something were to come up while your out and about, say running into a group of friends and jumping in their car going out for a bite (yes there’s COVID but it shouldn’t last forever)
As for the porch, 11 isn’t old for a mini pup (chihuahua). Our guys lasted a long long time, Judy was 17 when she started slowing down and couldn’t do the 4 miles around the lake, so we FrancIs. Judy lived to 22 years, Frannie lived til he was 25.
Since you stated the dog only needs to rest for about 20 minutes the purse seems the smarter choice, this way if you stop at a store et it’s easier to get about without the stroller. Your back should be fine for toting around ten pounds extra weight unless you already have back problems.
Just my thoughts.