Do not use K31. What a clumpy mess that makes in a KBG lawn.
I have had good luck seeding a TTTF and Per Rye blend in existing lawns. I really like the look of that blend and the TTTF is definitely more tolerant of dry conditions than KBG.
7x14 Stealth SE Enclosed with 48" side ramp
2013 Gravely Pro Turn 260 with Kohler 26.5 EFI
08 Toro Z400 48" deck, 21HP Kaw with EZ Vac twin bag
09 Exmark Turf Tracer HP 36"
21" Toro SR4 Super Recycler
Stihl KM90 Kombi w/ straight shaft trimmer & tiller heads
2 Husqvarna 125B Blowers
Turfco XT5 Aerator
Turfco LS-22 Overseeder
EcoLawn 200 Applicator
I can do all things through Him who gives me strength.