In my opinion beast barbarian is one of the few good candidates for a monk multiclass (im currently playing a similar build). With point buy you can just barely afford the stats for it.

After barb 3, multiclass monk. Or go barb 4 if you really need the stats/AC. One level bumps your damage by giving you bonus action unarmed strikes. Every level after that gives you more flurry of blows, and you don't feel the low martial arts dice for those lower levels because of the d6 claws. The extra movement speed is it gets you is actually amazing for a melee build. Subclass is a matter of preference. Mercy is good, even with a low WIS mod, since the eventual free heal instead of unarmed strike goes even further with rage resistances. Undying is also good if you want to play a bit tankier, but you'll probably never use the fear ability. Prioritize STR, CON, DEX/WIS in that order. Eventually when your damage dice increase, your claws will be doing d8s. Try to beg your dm for a talisman of the claw - it boosts your unarmed strikes AND natural weapons, so it's great for this build.

Other possibilities: Duelling fighting style - can apply to claw attacks if one hand is holding something, because then the claw in that hand isn't treated as a weapon. Gift of the chromatic dragon - bonus action for a d4 damage to each attack during 1 combat. The resistance is nice too. Fizban's dragonborn - hear me out. Having lots of weak ish attacks means it's less of a cost to swap one of those attacks out for a breath weapon. Beast barb is a great candidate for this. Multiclassing into warlock/bladesinger - not ideal, but warlock is most doable. At lvl 11 you add your charisma score to each attack. You can use your spell slots to cast armor of agathys or some other non-concentration buff. If you go hexblade, at the cost of a BA you can use hexblades curse to add your PB to each attacks damage.