I double the not picking her up - at this age, you may need to (taking out to potty, for example) - but eventually, you won't be able to, so it's not a big deal.
If you do pick her up and she squirms, just wait it out. Use a towel or blanket for protection, if you need it. Don't squeeze or hurt her, just wait for one split second of calm on her part before you put her down. And be sure you're not hurting her or making her uncomfortable with how you are picking her up. It might be worth asking the vet about it, to make sure there's not a medical issue making it hurt when you pick her up.
You will need to be able to handle her all over though - so I would work on some calming exercises. Just petting all over, massaging the feet, ears, tail, etc. You want to instill calm as you do this and make it a pleasant experience for her. There are some books on
TTouch that might help.
I would also suggest (just that, as I don't know all the details, so this is just a suggestion) that she might be over-stimulated and need more downtime and alone time in her crate (hopefully she has a crate or other "den" space that is her's alone). Having a shish tsu and goldendoodle in the house, I'm guessing there's a LOT of excitement in the house. She might just need a NAP.
Be calm, confident, and consistent.
When she bites, say "no" calmly, then stuff her mouth with a toy (plushy toys are good, or a rope toy). When she bites on the toy, say "yes!" in a happy voice.
Keep at it for another 4 months, and she should have it down! (Yes, it might take that long).
She's only been with you for 2 weeks and is still learning the rules - so be consistent with the rules, and she'll understand what's expected. Don't give up on her yet.
But - be realistic, too. There is a wide variety of personalities in the CC, but she might just not be an "in-your-face" kisser... and even if she does get to be that way, she might need some time to WANT to do that with these strange giant humans she now finds herself living with.
And be sure to get your wife on board - these are FAMILY dogs, and it will be important that everyone in the house is on the same page.
I'd also sign up for some puppy classes - you have other dogs, so you might not need the training aspect of the classes - but there's still benefits to getting out just you and the puppy for some bonding time and doing some socializing with other puppies in the same age group.
One last thought - check her food. If her tummy's upset over a new food or if she's sensitive to any wheat or grains in the food, that might be causing her extra stress and an/or unstable blood sugar which can lead to strange behavior. An upset tummy might also make picking her up uncomfortable... so just something to check.
And... we'd love to see some pictures of the pack, too!