My girls are now 12 and 9 but we have always used the UV swimsuits with long sleeves and arms. I even have photos of them on a beach on the Moray Firth in the north east of Scotland wearing their UV suits. neither of them has ever had sunburn because of these suits (we still use them every year even now) and it's so much less of a worry as to whether they have cream everywhere as there isn't a lot of skin exposed to the sun.

Edited to add: We have taken the girls all over France (including the south), Spain, Turkey and Florida and used these suits.

Sorry, having just reread my original post I realised I missed out what I actually wanted to know...blame baby brain!

My worry is if she'll be warm enough in the water with just the swim suit, or if I should be putting her in a wetsuit instead.

Even in Northern France mine were warm enough as babies in just the UV suits. The wetsuits don't look very comfortable for a baby as they aren't as soft and flexible as a UV suit. I wouldn't have rated my chances of actually getting one of mine into a wetsuit.

The wet suits will still allow the cold water to flood in so the initial shock will be the same so not really a help.
With ours we have put them in a sun suit and then put one of the little life jacket/buoyancy aids on, these fit nice and snug so keep the body core warm and provide some safety too.