Hi, I’m a plumber and bathroom/kitchen fitter. I specialise in bathrooms. There are many variables which can have a marked effect on pricing. What you have described sounds about right for an average bathroom. It may be a little lean. Don’t forget, you can pay a lot for a load of you know what. I’ve come across many fitters who claim to have been in the business for years and years and have been doing rubbish work for as long.
You nearly always get what you pay for. Ask for references and be sure to follow them up.
Incidentally, I only work on recommendation. I do no advertising at all and rely totally on word of mouth. That’s not because I’m cheap, it’s because I’m trusted. A large portion of my income is thanks to other fitters who have done poor work that needs to be remedied.
You need someone who has a good track record.
Words of advice for would be bathroom buyers. Never let a “mate” do you a deal!
All in all, you may be ok. Make sure to get a written quote with a lot of detail. Bathrooms take time and a fitter who has put in a low bid to get the work and later encounters problems will use up any wiggle room you leave open.