If money isn't an issue, then stay at one of the Universal hotels, I've heard they're nice, and you'll be right near City Walk, which my family really enjoys. They have evening "street performers" that are very cool and a huge TV screen that is pretty neat. The atmosphere of City Walk is nice for a while, and the restaurants are good too. Keep in mind, the three Universal hotels (when I last priced them) were $200+ per night. (As an add-on to your current vacation plans, it seems rather pricey to me.) Although, you could really enjoy yourself and only hang out at City Walk and the pools for a few days, because as far as visiting the Studios and Islands, your kids will not be able to participate in many of the attractions. We took my son and daughter when they were 4 and 15 months, and there wasn't much for either of them to do. It didn't really become interesting to them until last year when my son was 7 and daughter was 4 and finally tall enough to do most rides. They just don't have the same family friendly attractions the Disney does.
I would suggest skipping the Universal parks, based on your kids' ages (it would be pricey for not much entertainment, in my opinion) but staying in the hotels might just be really cool. I have not stayed in the hotels, so I can't say how nice they might be, but they are relatively new, so I imagine they are in good shape and for the price, rather nice. There are plenty of other nice hotels/resorts in the area with cool pools, maybe it would behoove you to stay at one of those for the few days before staying on site at Disney. We've stayed at Sheritan's Vistana (you can see it from I-4, I believe it's right off of Apopka Vineland) and it was very nice. I believe it's priced less than the Universal hotels and you'd be closer to Disney (and closer to Downtown Disney too.) We've also stayed at Cypruss Pointe Grande Villas and rented a 3 bedroom villa (less than $170 per day) and it was awesome. (it's located on Hwy. 525, I think. Anyway, we were just 4 blocks from the entrance to Disney property, near Downtown Disney). They have a cool volcano pool, so your kids might enjoy that. Anyway, just thought I'd suggest another option to Universal, and if swimming is a must, find a hotel with a cool pool!!!
I hope that helps! Good luck and let us know what you decide. :wave:
Edit: I wrote this reading that you can't get into ANY Disney resorts (which I don't think I read it correctly.) I think ANY Disney resort trumps the off site resorts (if money is no object and you don't mind sharing a single room with your entire family), so I'd take whatever Disney room is available, however if that isn't possible, then I'd consider Universal as well as some of the surrounding resorts. This is coming from a person who has never stayed on Disney Property. Our preference is to have a 3 bedroom rental condo or house, which has always offered more room for our family, at a much lower price. If I could get the same ammenities for the same price on Disney property, my life would be perfect!!!!!