Width doesn't matter as much as tread and rolling resistance. Get something with a small tread that rolls fast. Continental speed king/Schwalbe Thuder Burt, or any gravel tires that are at least 40mm wide should fit on your 29" wheels.

I would try with the current tires at a higher pressure such as 2kg and see how it goes. If you can ride with them, nothing to be done.

Continental GP5000 clincher tires 700x23.

"29er" is the same thing as 700c, so as long as you choose a tire that's a good bit wider than your rim's inner bed, you should be safe running 700c gravel tires.

What's your gravel riding like?
What sorts of surfaces? Is there a lot of mud, or is it mostly rocky aggregate? How chunky does it get: are you looking at smooth hardpack with only small bits of rock, or is it loaded with 2" chunk or whatever?
What kinds of gradients do you deal with? Is it mountainous? How much flat-ish stuff is mixed in? Do your friends prefer mixed-surface riding with lots of pavement in the mix?
Have you tried just going on a gravel ride with your bike as-is to scope out the situation?

If you'll be dealing with seriously chunky gravel, you might not want to downsize your tires much (if at all), especially since it'll modify your bike's handling. But even if you stick with fairly wide tires, it might make sense to get a more performance-oriented tire, if you're currently running something pretty beefy. It just depends.

I raced Midsouth (Landrun) 100 on my MTB this year with 700x40 Conti Terra Speed tires. I did it because i knew there would be mud and went for max tire clearance on a MTB vs the better aerodynamics of my gravel bike. It worked fine, but the conditions were wet enough that the gravel bike would have been a better option. Anyway...

Running a gravel tire on your MTB should work fine. I'd say for gravel it's a "marginal gain" over just running a fast MTB tire. But if you decide to run that gravel tire on a MTB trail it would be a significant setback and you'd have a higher chance of damaging your rims and tires.

If i were in your shoes I'd just get some fast 2.1-2.3 MTB tires (like conti race king protection) and ride them on gravel and on MTB trails.

If you have a suspension lockout you'll be in good shape. Slap some aerobars on your MTB bars and you'll be even better.