Yes: Do a lust build, just tease. Don't need a weapon to tease. Do note that you'll still get severely fucked up by anything immune to teases (like mining robots) but tease is ridiculously strong against many opponents. And also there are passive benefits to weapons you'll also miss out on. You'll never be as effective as a build which actually uses weapons, or even just has them equipped, but that doesn't matter when most foes get shredded.
Also I don't quite get what you mean by "still counts though." If you mean you count them as weapons, then like, you'll have absolutely no chance against a mining robot or something like that, but if you mean that they count as the "anything," then you can try spamming emp grenades to deal with them.
For classes, Smugglers get high evasion and also a few non-item grenades for use as abilities (like the gas grenade, which deals lust damage but honestly isn't very good beyond the group damage, given how tease works after the rework). If you insist on doing physical, The evasion, along with effects like the flash grenade and stealth field, give you high potential survivability, and at level 5 you can get the normal grenade rather than the gas grenades and use that, and though other classes are probably better for single-target ability damage, it is a group attack.
The other two classes, I'll largely be basing it on the assumption of not doing a tease build, because of their abilities. That said, if you are going for a tease build, the Tech Specialist's heavy tankiness is very useful.
The mercenary is largely weapon-based and damage dealing, with decent tankiness, and is a bit unique compared to the other two due to the level 7 perk boosting willpower allowing for slightly more tankiness against lust enemies. Headbutt is an attack that deals damage and explicitly ignores your weapon's damage, plus the stun chance, despite being unreliable, can be helpful. Level 4 is good survivability too, either allowing for constant but relatively low evasion as you punch or a periodic ability to nullify ranged damage. Level 5's grenade/charge choice is useful for either group or single target damage, whichever you may think you need. After that point it's still pretty decent, but like, it won't do that well because unarmed playthroughs are not intended and the grenade/charge abilities fall off terribly at high levels.
The tech Specialist, meanwhile, is very tanky due to their automatic perks, and also is THE BEST in terms of single-target damage. Why? It's because Charge Weapon works with the rock (your unarmed weapon). Charge Weapon is a frankly overpowered ability due to it providing high levels of damage on top of your normal weapon's stuff, so even if you're not using an actual weapon you'll still be doing good damage. That's at level 3, of course, but earlier levels are good too: Paralyzing Shock is paralysis, which definitely boosts your survivability when it works, and at level 2 the attack drone is nice bonus damage. The Disruptors are very good single-target damage, and though they fall off at high levels compared to weapon attacks+Charge Weapon, they'll be quite useful with no actual weapons. The level 6 perk selection is useless and the level 7 perk selection is also not that good, for this style, since you don't get an energy weapon. That said, the blind/damage effect of Charge Shield can be decent, and Shield Hack becomes more useful when you don't have a weapon to actually deal damage with (though Charge Weapon still rather nullifies the need for it, since it does electric damage). 8 is amazing like it is on every class, since all classes get the same auto-perks and perk selections at that level. At level 9 I'd recommend Drone Control since it buffs your Attack Drone (or Tamwolf/Siegwulfe if you're using them. If Tamwolf, I'd recommend taking Attack Drone at level 2, since even though equipping it weakens your health pool, having that perk gives you the repaired version when you first acquire him, and lets you upgrade him to a superior version, in addition to having the damage buff from the attack drone perk. If using Siegwulfe, however, I'd recommend not using the Attack Drone perk, as the damage bonus is not too great compared to the larger shield you'll have otherwise, and unlike Tamwolf she comes in her base state and can be "upgraded" regardless of if you have this perk), while Deadly Shock's damage is just on the use of an ability that should be unneeded at this point.