I tell my engineers - if you get paged off hours, however much time you spend resolving the issue, take that time back from the next workday.
I also practice what I preach - if I get paged at 3am and work on the issue until 5am, I’ll come in 2 hours late or leave 2 hours early.
It sounds like you’ve got a good manager, so hopefully they won’t hold that against you. This is the reality of oncall - it sucks!
When you get woken up in the middle of the night, of course you’re going to be more tired the next day. I’m the same as you - I can’t fall back asleep if its early morning so I normally just stay awake and am tired that day. You shouldn’t feel guilty for being at 1/2 capacity after working all night to solve your employer’s problems!