Hi, and welcome. Nice you are interested in giving a baby a forever home. Was wondering if you heard back.
If money is the object with grooming, bathing is no big deal at all. You can make a shampoo solution and gently smoothe it in in the direction that the hair grows. No need for scrubbing or rubbing, then rinse thoroughly. Conditioner, if you think the hair needs it but always rinse completely. I blow dry my kids, and they are very used to the whole routine now.
If you like puppy cut, take your baby for clipping or learn to puppy cut the baby yourself. If you choose the groomer, but want to save, have the cut shorter in Spring, then let it grow out and just get nails clipped, if you are nervous about doing that yourself.
I keep long coats, trim and round paws and clip ears and bums myself, and get nail clips as needed. I can clip nails, but with three, it takes me longer than the groomer, including the drive...and the groomer rate for nail clips is very reasonable, and she takes walk-ins for nail clips. Plus, it gives my kids a trip out...LOL. Your vet may be an option for nails, too.
I wouldn't worry about the breed/mix but hair could be different in a mix. My kids are silky coated Yorkies but if they matted easily, I would probably clip shorter for their comfort. The great part is you get to choose as often as you like.
If you get a Yorkie/Shih Tzu mix, you could have thicker and slightly different texture hair. It shouldn't be too big a deal, though...you have plenty of options. If a Yorkie/Chi mix, you could end up with a smooth coat and no need for hair clips anyway...wash and wear hair...LOL
God luck...let us know!