I did see one woman in a bikini at the Diplo, but admittedly she was on the beach, not by the pool. Plenty of "tankinis" about...more covering version of the bikini.

None of the big hotels are taking membership requests till after Eid - you could try the Intercon at that point. You can also visit the Intercon as a non-member during the week (not Fri/Sat) to see what it's like, though it's pricey.

Oasis is next to Sharq village, past Doha port and near Doha Sailing Club - keep heading up the Corniche with the sea on your left, you'll find it. Their number is in Marhaba.