First off, welcome to the forums
For that budget you should be able to get something decent (and I don't care if people say "no, you must drop £800+ into a boltie to make it good" because they're objectively wrong

). Patrolbase has both the JG BAR10 (
https://www.patrolbase.co.uk/jing-gong-bar-10) and the Cyma cm.701b in stock at the moment (
https://www.patrolbase.co.uk/cyma-cm-701b-vsr-bar-10), both of which are based on the TM VSR platform, which has been a really strong platform for over a decade, so there's an absolute plethora of parts. The JG comes in at £120 and the Cyma at £130, so that'll leave you with room for upgrades.
For immediate upgrades to put into it, I'd look at the Action Army hop unit, a 70 degree silicone Maple Leaf Autobot bucking and a Maple Leaf Omega nub. This'll put you up to just under £200 and the rifle will perform decently well. Some sort of cheap scope if you don't already have one will be useful too; you can get plenty of decent ones for around £50. Then I'd pick up two spare mags (£10 each) and call it a day for now. You can stash the mags in a pistol mag pouch, or even just in your pockets.
A tightbore barrel would be the next thing I'd get, which'd put it up to about £320ish (depending how much you spend on the scope), but this isn't really a necessity as the stock barrel will do the job well enough. My choice for barrel is Laylax because I'm old school, but I know people love the Maple Leaf Crazy Jets too.
Then power upgrades cost a lot and don't make a huge difference from the stock 420fps/1.7ish joules power, so I'd just use it until you decide you want to invest more, or until the internal parts break.
The other thing I'd ask is, do you have a sidearm? Having a pistol is pretty essential when you're running a BASR, since you'll have an MED to deal with, and BASRs aren't the greatest platform for putting down any weight of fire in case you get caught out while moving. If not, I'd actually get a pistol first and run it alongside your AEG. Make a conscious effort to use it, since with an AEG you rarely actually need a pistol, but it'll help you get used to the movements and shooting it, which will help when you transition to using a BASR. For sidearms, I'll always recommend TM pistols, because they're fantastic and work all year round, though they do come with a ~£150 price tag. Good thing is, you don't need to do anything to them; they shoot fantastically well out of the box. Yes, you can run a BASR and an AEG (like your Specna carbine), but unless you practice doing it, it's awkward and clunky.
I just realised I've written quite a lot, so I'll leave it there before I ramble even more. I love sniping in airsoft and could go on about it for hours ?