He's a twat!
My ex was army and did a 6 month deployment when dd was just over a year old so yes SOME mothers do have to do a lot more in terms of childcare BUT as the first respondent rightly pointed out they're ALSO not burdened by the extra work of another adults needs.
In addition if needed army wives can ask for support from the welfare officer who can make certain arrangements. It was less formalised in the past, but I believe now there is essentially "sure start" type support on most bases. In my mothers day the other wives attached to the unit would "pull together" to help each other, from fetching shopping or hanging washing to babysitting - so they didn't have to do it all themselves though admittedly most would try as a matter of pride and other wives would step in where there was illness, bereavement etc to deal with.
When ex was home we did similar to you, I went to bed "early" to get a few hours in before first night feed, went to bed himself whenever I was going that feed, then up at 6/7 to get ready for work and he'd exercise using dd as a "weight" or else just making silly faces or singing to her while he got ready, then he'd fetch me before he left for work.
My dad & grandad did similar (both also army)
So as he seems to be comparing you to army wives HE can act like an army husband and not be a lazy unsupportive arse!!
"I can see both sides. It’s his child so he should help out but if he is working full time his sleep is important or he can’t function at work" hate this argument it's bollocks! BOTH parents need as much sleep as possible. One for working outside the home, one for working in it. The sahp also needs their wits about them to keep everyone safe & healthy. A dead on their feet sahm carrying a baby down stairs is NOT ideal and should be avoided if at all possible.
Bro is a police officer - also pulls his weight with DC and indeed at one point was a single rp dad! He'd rip the piss out of your dh!!