Yeah so I think the best solution is just to buff it rather then then change the pop since its much easier to fine tune % stat change then effectively just doubling it per pop.
IIRC their defenders aura is final stat based so if we just introduce small changes to buffs it will scale pretty well in the end game.
So just off the top of my head about the sentinel upgrades (not accounting for the HP bonus) it gets about 2.8x stats from all of its upgrades and then the final defenders bonus which stacks multiplicatively so its about 3.22x stats at the end. This gives it an attack per pop about 30% short of a late game Janissary, which is already pretty lacklustre.
If we just give it another 20% atk buff somewhere and also increased the defenders bonus(either from buffing wall guns, the combat card or anything) then it would basically be the same as a Janissary which combined with its other advantages I think should be enough
edit: so just to list out the suggestions
- Combat card to 20% stat
- wall guns to 20% atk ( and allowing all defensive buildings to affect it with increased range)
- and battle bluster to 50%
- defenders bonus to cap to 25%
edit edit: forgot about the gunpoweder depot but that is risky so lets not count it and math was a bit off