Private rooms should have linens, from what I remember. Towels, probably.
But I'd still say: YES, bring a lightweight (!) sleeping bag.
A lightweight sleeping bag only weighs ~500g. A small microfibre towel or dish towel maybe 100-200g. I'd consider those essentials for a Camino, unless you have pre-booked all your private rooms or you're happy to take a taxi to the next hotel if needed.
Even if you carry towel and sleeping bag, you can still keep the weight down easily. Less clothes, less electronics, less cosmetic items...
April can be cold. If for whatever reason you end up in an albergue dormitory at some point, instead of a private room, and there are no blankets available (not that rare), you'll probably wish you brought a sleeping bag...
Personally, I'd prefer to carry 500g more over the risk of being cold and uncomfortable. But that's a personal choice.
Happy planning, Buen Camino, and merry Christmas!