I'll be honest with you, OP. I think you should, if at all possible, push the budget up slightly to at least £250-£300. At Grade 4/5, you have to learn a lot of new techniques, and £150 (unless you're very very lucky and get something second hand from another student) is only going to get you something cheap, that sounds like it.
Two bits of advice: I know I've said this before but please do contact specialist string shops. I mentioned Cardiff (who are terrific), but there are others (e.g. Woodbridge) around the country. Call them for advice. I find them usually incredibly helpful - after all, they have a vested interest in supporting the next generation of musicians (and customers). The instruments that a specialist shop sells will be properly set up, and there will be no need to pay extra on top.
Also, while it's of course best to actually go to a shop with your DD, that isn't always necessary. I bought my DD's fantastic 3/4 violin after having several conversations on the phone and via email. The violin was couriered on an "approval" basis, arrived in pristine condition and the whole process couldn't have been easier. And when it comes time (soon, I think) to move up to a full-size violin, I can trade her current violin in.
Two ways to possibly reduce costs: do ask about second hand instruments. All decent string shops offer generous trade-in policies as students progress to bigger and better instruments, and these instruments (which have had a once over in their workshop to make sure everything is tip top) can offer good value - about 25% off or possibly more. So do ask if there's anything used.
The other thing is the Assisted Instrument Purchase Scheme, which you might be eligible for (if your DD goes to a state school and either 1) learns her instrument or 2) plays in an orchestra within the local music service. AIPS basically saves you the VAT of 20%.
Anyway, I hope the above isn't too much. I've been there, that's all.