I could help find something suitable, but have some questions.
What's the boy's height? Is he actually going to use the thing or get bored of trying and let it rot in the garden? (his friends got bikes bmxs already and they have somewhere to ride them). Often kids just give up quick and end up hanging around on scooters instead. Is second hand out of the equation as it's a present?
Bmx completes and parts have doubled in cost in the past decade, the bikes are easier to learn on but some parts haven't changed for the better. £300 will not get anything good or will last without maintenance if he uses it and/or doesn't look after it. All the sub £500 completes are the same, bar a few exceptions. You'll be lucky to get any money back if he decides he doesn't like it, unless locally there's someone on FB marketplace that takes an eye to it, the market is very dry at the moment.
I've had a fair browse and found this, few colours to choose from and 20.5" being the ideal size unless he's super small. Even if he's 6' tall 20.5" is fine, it's got a medium rear end and low bb.
https://www.skatepro.uk/56-42706.htm
It follows the same trend as all the other sub £500 bikes. It does sacrifice a cartridge bearing headset which isn't an issue (if in the case kid chooses to upgrade bike over time he would be limited to non integrated-race forks, which is easily overcome by buying some second hand from a previous gen, which are stronger anyway), and fully sealed rear hub, but does offer sealed driver bearings so maintenance won't be overly annoying. Unsealed bearings are better than cartridge if the cartridge ones are cheap. The steerer tube diameter does look narrow on the pictures, however all the hi-tensile forks will bend with ease anyway, we always rode bent forks when we were younger. Key selling point being, bike will suffice until he reaches a point where he's ready for something better and it hits under budget from a reliable store. It's just if he likes the look of it too.
The Sunday Blueprint bike in the earlier post looks very nice, but it falls into the sub £500 'same as other bikes' category when you break it down. Many of the brands are good at making their complete bikes line nice looking, but they don't deliver. Head to head with the Haro there's nothing really in it.
), although it appears bars are now generally 1.5-2" taller than they used to be, which also plays in to the geometry.