Getting from Terminal 4 to Terminal 5 at Heathrow Airport Early on a Sunday Morning

Bus H30- which is the inter terminal night service when the tubes and trains are closed- https://www.diamondbuses.com/bus-services/se/seh30/

Your hotel can also call one of the local mini-cabs if you prefer the convenience of door to door service with your luggage.
When we stayed at the nearby Premier Inn at T4 the cost to get us to T5 via mini-cab was about 15 pounds, fyi.

If it's long-haul and you need 3 hours, taxi or H30 bus.

If it's short-hail or UK you need 2 hours, taxi or 490 or 482 bus. You could also do a shorter bus run to Hatton Cross and take the Underground from there as that part of the line does run that early on a Sunday, but not the T4 loop.

That all said, I would look at changing hotels to T5.

Even for a long haul flight the first bus 490 is at 0350 and the first 482 is at 0353- both only take 12 minutes between terminals, so going via Hatton Cross (in the opposite direction on the bus) for the tube takes more time.
Those are all day buses.
The funny thing is that the H30 is a free bus whereas the 490 and 482 are normal London bus fares of £1.75.
There is even a National Express coach #A30 between the two terminals at 0327 which can actually be booked between the terminals.
For me, with a Coachcard, NatEx is actually cheaper than the London Bus routes. Possibly worth doing sometime just for being maybe the silliest NatEx journey you can do.

What is the best way to get to T5 at that time of day?

It depends on what time you want to get to the terminal, whether or not you have bags to check and if you will have your boarding pass prior to getting to the terminal.

The last H30 bus leaves T4 at 4:37 and gets to T5 13 minutes later. Do you need to be at the terminal two hours and 40 minutes prior to your flight? I doubt it on a Sunday morning.

The 482 and 490 buses run every 30 minutes on a Sunday morning.

So if you want to take the bus route you need to watch your times carefully. Miss a bus and you have a long wait.

Personally, I would ask at your hotel. You're not the first person to need to do this and they might have a better answer. (Possibly a deal with a local cab company.) And even if they didn't have a deal, I'd have them arrange a cab. Sometimes, convenience is more important than saving a few pounds. Especially at 5 AM.

You could consider switching to the Sofitel hotel adjacent to T5 for your Saturday night stay. It's a very nice hotel, doesn't appear to be that much more expensive than the HI, and is a short walk to the check-in area for your early departure. Would certainly ease your logistics somewhat.
For our own early departure last year we were able to walk over to the check-in desk/kiosk the evening before and collect our boarding passes, which made for no-stress and seamless processing the next morning.

Comply with what your airline tells you is your required check in time, not with what some stranger on a forum or a hotel tells you they think will suffice.
You've been given all the options of transportation now. It's up to you as to which you consider to be the "best"