Update: After trying yet again on both the front and back (6 tubes total)...my bike shop determined that the tires (lined with kevlar) had too wide a bead and were too thick for that specific rim.
It's not something I would've been able to tell. Not something they even could tell right away. The tube was sneaking into spaces it shouldn't and was getting pinched or pulled.
Hopefully, my misfortune (I'm out a lot of money) helps someone else. I might sell them on eBay or TE. They're hella cool tires. Green sidewalls with a black racing stripe. 700 x 28C.
Welp. I'm glad the old girl is on her feet again at least.
"Things look different from the seat of a bike carrying a sleeping bag with a cold beer tucked inside." ~Jim Malusa
2009 Trek 520-Brooks B-17 Special in Antique Brown
2010 Surly Long Haul Trucker-Brooks B-17 Standard in Black
1983 Fuji Espree Single Speed-Brooks B17 British Racing Green