I did some digging and looked through CT portal.ct.gov and found nothing that said they can't. However, as stated by portal.ct.gov, it costs either $30 or $70 to do it it via fingerprints Ing by the gov or through a private company respectively. There is also a free version, through the state for DDS reasons. I am not familiar with the acronym, sorry.

My opinion is the same as yours, do not bother. If a company is going to charge you to work for them, they are not a company worth working for. Especially for less than $100. That's a big red flag.