he's fussy within an hour or so of waking up.
Yes, he's tired. It's normal for a baby this age to be ready to sleep within an hour of waking.
He has approx 2 hours of awake time before showing signs, eye rubbing etc
Outward signs like eye rubbing etc are signed baby is now over tired and needed to be fully asleep at least half an hour earlier, probably more.
Have you seen this thread? Baby is the same age as yours.
www.mumsnet.com/Talk/sleep/4174156-5-month-old-tired-and-sleeps-alot
I mean thus gently, but I think you have your expectations wrong. A couple of things, for example ...by which point he only has an hour or so before waking to eat.
Yes, that's right. Really life with a young baby is mostly just
wake
feed
A little bit of floor time until grumpy, an hour would be a lot, more likely 30-45 mins or so
Settling to sleep
Sleep
and repeat.
There isn't expected to be huge chunks of time to "entertain" then when little. This struck me from your OP too, so much so I was going to add it yo my earlier post but decided it was irrelevant
As a newborn he was so fussy, if he wasn't sleeping or eating he was crying / fussing
See - newborn literally only eat and sleep. So your baby wasn't fussy. In those first few weeks literally life is just wake > cry > feed > back to sleep and repeat over and over again.
If you got into the mindset that baby needed entertaining out of being upset, rather than settling to sleep (which is understandable, many FTM don't realise the amount of sleep babies need) then u can see how it now would seem very strange to try and help your baby sleep much more.
Speaking as both a mum of four (very different babies) and also as a sleep consultant, I'm absolutely certain that the primary issue to tackle here is not enough sleep over 24h.