Macro flash ring for Nikon - but on li-ion batteries?

I'm sure it would be a whole lot cheaper to just buy a stack of rechargeable NiMh AA batteries & carry spares with you.

Li-ion still need recharging, even if they run better overall than NiMh. You can get Li-ion rechargeables in small forms like AA & AAA, but they are considerably more expensive. UK Amazon prices as example - AA NiMh, 16 for £25 (I got a free 8-port USB-powered charger with mine), Li-ion, 4 for £25.

Pick a supplier who provides tough plastic boxes, so you don't have to carry them loose.

There's one additional factor, which may or may not affect you. Aircraft flight operators have varying rules on being able to take Li-ion batteries on aircraft. Some are tougher than others, but be prepared that you may not be able to fly with them "outside of their device". I've previously managed to negotiate that a "device" can be the charger.

BTW Google found several Li-ion ring flashes for a mere £500 each, even for 'cheaper' manufacturers like Godox… for which you'd need to buy at least one additional battery. I have a pair of regular Godox strobes that run on large, custom sized, Li-ion batteries. The strobes were 300 each, the spare batteries another 50 each.
Everything else, I run on NiMh AA or AAA.