so between the weld at the bottom for the skirt and the first rib is about 1"? that should just about fit a 1/2" weldless kit. you'll need a dip tube of some kind on the inside to get all the liquid out of the kettle.
bulkhead + interior dip tube:
ball valve for the outside of the bulkhead:
hose barb for the end of the ball valve:
put a couple feet of silicone 1/2" ID tube in there and you're home free.
as for the volumes, where are you getting these recipes? from a kit? every credible source of information you'll find says you want to boil the full volume in the kettle, not top up with water afterwards. you'll get better utilization of hops and ensure you've got "clean" water since it's all boiled. depending on your burner and environment, you'll boil off about a gallon in an hour boil. accounting for hop/trub losses you're already near like 6.5 gallons. the kettle won't really hold a full 7.75 once you cut the top off it near the outside handle (easier to work with and clean). that puts you at like 7.5 gallons with 6 - 6.5 gallons in the vessel. I'm not saying it can't be done, but you'll have to watch the boil REALLY carefully for boil overs, especially at the start of the boil.
for the same amount of effort on a full 1/2bbl keg, you can up your production and head space. but if you're ok with the 1/4bbl then your plan is solid