Yes. You need to have your name on the title to exempt (protect) it.
In Utah, we cannot exempt (protect) your interest in the car unless your name is on title, so you will most likely need to get your name on title quickly. Under our current exemptions, I can protect $3,000 of equity in a vehicle for husband and another $3,000 of equity in a vehicle for wife, so long as both names are on title.
Buying a car with your tax refund is generally a good idea, unless you go over the exemption limit or have too many vehicles (meaning we cannot protect the recently purchased car).
Recently buying a car can adversely affect your bankruptcy, but only if your income is high enough that the trustee is worried that you purchased the new vehicle in order to drive up your expenses.
You really should talk to an attorney and plan out what to do with the tax refund and the car. Since you cannot protect your tax refund under Utah bankruptcy law, you’ll need to spend it all on exempt items, and that’s something you’ll need a long discussion with an attorney to plan out in advance.