with that kind of budget, I'd choose a combo of rear RE001/T1R/whatever tires in the range of RM250+, and Toyo R1Rs for fronts.
You can run a maximum of 205/50R15, but hard to find those tires. RE001 have those sizes, and they're only made in japan, can get, but no shops stock it AFAIK, and the only way is to order beforehand. RE001 in 195/55R15 and 195/50R15 are made in Thailand. Toyo, Recently ordered (4-5months ago, I think) and can get 205/50R15.
205/50R15 and 195/55R15 almost the same size in height, the only difference is the extra 10mm of width.. looks wide when installed on a civic 5th or 5th gen.
Sedan cars, don't use grippy tires at the back... you want a bit of give when doing sharp corners. If using both grippy tires, need to run harder suspension to get the benefit of faster cornering speed. If not, welcome to world of massive understeer.
And if you're that particular about grip, get an LSD. That makes a difference even on standard tires, esp. during wet weather.. nothing is more annoying than having one tyre slipping when going over puddles.