Without a price, it's really difficult - but then so is trying to keep baby in the frame with aunts and others creating mayhem when all you want is a steady shot. Oh and babies sleep a lot and only play on the floor, so a good set of knees will be needed - and/or a low-level holding device, so you can film at baby's level.
Lighting indoors can be tricky - in the past it was all but impossible without plenty of extra lighting, but modern camcorders are surprisingly good - and you can increase the lighting in the room quite easily by changing bulbs to the next higher rating. I would suggest separate lights are likely to be brighter, although the convenience of something built-in may appeal it will severely restrict yr choice. Also, you may discover it's not strong enough and it will load the battery for sure. There are many multi-LED panels available from about £30 and these should outshine anything "Built-in" Also note that I read here that at least one camcorder shines the light very close to the lens - so it is inside the filter ring....with the prospect that will induce flare. While close to the lens is otherwise "good" - Ideally a ring-light ( surrounding the lens), these are almost absent from Mfrs accessories - and those that are available are fiendishly expensive. In short, you probably don't need a built-in light which is why few come with them.
+++ We really need to have that Budget . . . .