Stay away from anything organic or that can absorb water or you will have mold issues in the future. LVT is a nice way to go providing the floor is flat and or level. You can always use area rugs to soften the look and deaden the sound.

Coretec LVP has cork backing, will help absorb noise.

You won't get much sound absorption from the floor (floor is more of a sound transmitted to the people below you issue)

But you can put a sound absorbing "high mass vinyl" sheeting under it before installation.

Anything that breaks up solid planes will minimize sound - upholstered furniture, rugs, bookcases, hanging textiles - by minimizing the bounce.

Cork floating floor + 6mm cork underlay = 20 dB worth of sound absorption. And I would argue that a floor that absorbs sound will reduce noise transfer "up". Cork "eats" airborne noise. The airborne noise is the stuff that goes "up" into a ceiling or down a hallway or up a set of stairs or through a bedroom door. If you kill the airborne noise, you kill the transfer to the other areas. It is that simple.

I've had hundreds of people use cork floating flooring in basements (no issue with mold...because you need a vapour barrier = same as a vinyl or laminate floating floor) and they have all been ASTOUNDED by how QUIET the space becomes. And they also comment on how much the cork reduces the noise from transferring 'up'.

Vinyl flooring = non-absorptive. The mass-loaded vinyl (underpad) is the stuff that reduces noise...but it will be as expensive as the vinyl floor itself.

If they can't afford a polished concrete floor (starts at $5/sf) then they are going to have a VERY hard time finding absorptive material. An acoustic "retrofit" is one of the MOST EXPENSIVE projects we have in the building industry. I estimate an acoustic retrofit to be $10/sf. If you can get it BELOW $6/sf, you are LAUGHING!!!

iCork Floor has cork floating floors on sale for $1.99/sf. The 6mm cork underlay (which does 80% of the acoustic work) is $0.66/sf. For $2.65/sf + labor, you will have created an acoustically insulated space. Even if your labor is the same cost...for less than $5.50/sf, you will be LESS than the BEST priced acoustic retrofit. Not something to sneeze at!

https://www.icorkfloor.com/store/product-category/thickness/floating-cork-flooring-10mm/

https://www.icorkfloor.com/store/floor-underlayment-cork-6mm/

Yes. The Golden Beach isn't the prettiest thing on earth...but it is just as effective as the rest. The 10mm Desert Arable is very pretty at $2.69/sf...it is still an acoustic option for low-budget clients.

The acoustic requirement with a low budget client is what will kill this project. It is either budget friendly or noise-friendly. I won't be both. Better let them down gently. And feel free to mention the $10/sf cost for an acoustic retrofit at a later date. That will be the part that will encourage them to open up the wallet a little bit more to get a cork floor installed - with cork underlay.