My daughter has CIRS and yes, that can cause tics. In her case, the mold exposure was/is in the schools. We found three things helpful - improving the environment, treating a fungal infection in her sinuses (frequently accompanies an infection called MARCoNs) and helping the body detox. I could write a novel on all of this, so I'll only focus on the environment since that's what you asked about.
We did some basic cleaning up of our home (replaced carpets with hardwood, tossed lots of things from the basement, run a dehumidifier, got rid of house plants) but our ERMI test showed that the house wasn't the problem. And there wasn't much we could do about the schools. So we fought a long battle and eventually got the district to put IQ Air GC Multigas air filters in each of my daughter's classrooms. Her attendance went from a few hours a week to full time. So definitely consider this kind of filter - it's portable and most sellers give you a 60 day return policy if you find it doesn't help.
For your house, if the problem is with the AC, then it's likely that mold spores have been spread throughout the house. So it's not crazy that the 2nd company is suggesting such an intense approach. But...I'm wasn't born wealthy, so I can understand wanting a more realistic approach. The AC system obviously has to be fixed in a way that the problem doesn't come back. And I would definitely ditch carpets (even if it means having bare subfloors for a time) - make sure the carpets get rolled up into heavy mil plastic and removed out of the house through the closest windows, not carried through the rest of the house. Remediate what makes sense in the bedroom, and clean everything you can on your own in the rest of the house (window treatments, all clothes, wipe down everything you can with Benefect or EC3 cleaning supplies). Get the IQ Air filter for your daughter's room and maybe the family room. Consider mattress encasements and replace pillows. Make sure your washer doesn't have mold. Do the easy/cheap/obvious stuff yourself. Then between this level of remediation and detox, see how she feels. If the tics start to subside and she starts to feel better, you can chip away at the problem until she gets healthy.
If she doesn't recover fully, then keep looking for a mold source and consider more extreme measures. You can find helpful information here http://moldcontrolonabudget.com/ The site is run by May Dooley and she's super helpful. You may ultimately need to go full out, but nothing says you need to do it as a first step. Hope this helps.