Can I just get a check from doctor 1 to pay doctor 2?
You are probably legally entitled to repayment of any advanced and unearned payments you've made to dentist 1. i.e. The payments you've made for services yet to be rendered.
Or is there going to be problem refunding this money?
I'd imagine dentist 1 won't be terribly happy that you want to switch providers in the middle of long term treatment and will likely make this process difficult. I suspect yes, there will be a problem.
Separately, since these are FSA funds and you're spanning tax years there's likely a tax issue in play. Assuming you can talk the dentist in to a refund, and they refund your FSA rather than giving you a check, your course of action depends on how your employer arranged the FSA. The employer can either allow a rollover of unused funds or an extended grace period to submit claims for the prior year. If they refund the FSA, that goes back to your employer and you either:
Rollover Up to $500 of unused FSA funds can be rolled over from one tax year to the next. You'd be able to make a 2018 claim against these funds for dentist 2.
Extended Grace Period, You have until March 15 of the following year to incur qualified expenses (you can use 2017 FSA funds for services up until March 15, 2018) and then you have until March 31 to submit a claim. But March 15 has passed and dentist 2 likely hasn't given you an invoice with a date of service on it yet so these funds would be forfeited; unless you have other qualified expenses that you haven't claimed against yet.
Ideally, you'd convince dentist 1 to give you a check, I don't know what, if any, tax impact there would be to receiving this check. I'd suspect the IRS would want to treat this as income since it would otherwise have been forfeited and wasn't used for qualified expenses (you can't use 2016 FSA funds for a 2018 date of service).