Should I wake my husband up at 5am to take over with our sick kid?

Our youngest has a stomach bug, so I'm on the night shift with her. I know from past experiences that I'm better off just staying awake rather than trying to sleep only to be awake again every 20 minutes. It's 3am now. My husband and I both work tomorrow, but both from home (which is convenient given the kids can't go to school tomorrow if one is throwing up).

He's been asleep since 11. I'm wondering if I should wake him up at 5 and ask him to take over so I can try for a nap before work or if it's better that at least one of us get a full night's sleep. Normally he gets up at 8, so that's three hours less sleep than he usually gets. Honestly, I don't think a nap is really going to fix how tired I'll be tomorrow, and it seems shitty to rob him of 3 hours of sleep just because I'm tired. Maybe I should just tough it out all night. But also I'm so tired right now I feel desperate to sleep. I'm too tired to make good decisions. Help.

Update: My girl continued to throw up until 5am, when I did decide to turn it over to him. I told him to stay in bed and just get up if he heard her again, but of course he went straight to her room anyway to be sure she wouldn't be alone if she woke up again. Ironically, he says she only woke up to throw up once with him. I wish I had toughed it out a little longer, knowing that. But how could I have known, right? And he's such a great dad and husband, he was more than happy to take over (and had me to know that I could have woken him up a lot sooner). I took an extra couple of hours off work this morning, so I was able to get 3.5 solid hours of sleep. I'm actually feeling pretty good.