Aeodynamics from early 1930s certainly put the handbrake on either of those Fairey or Hawker aircraft. An 1-engined fast British bomber needs to be developed by either Supermarine or De Havilland - the companies that knew how to make racers.
Racers rarely make good combat aircraft. I am not sure that some companies had the 'secrets' to speed. In a lot of cases the companies were trying to satisfy bunch of conflicting requirements. And the British Air ministry could sometimes (more than sometimes) ask for
more than a liter to be put in a 1/5 liter bottle.
DH Don. 250 ordered (off the drawing board?) but after the prototype flew and some trials the order was cut to 50 and 20 of them were 'finished' as ground instruction airframe without engines. Maybe not the guys you want to design your high speed bomber? or maybe just the guys if you can keep the Air Ministry guys from adding a ton of "nice to have features but are now part of the design or no orders"
A lot depends on the expected range? If the target is just a short hop behind the frontline, fuel can be loaded to 50% and more bombs can be carried. If the target is a few hundreds of miles away, keep just the internal bomb.
Yes, somebody should have figured out that a single engine bomber was not going to work for strategic bombing (1000 mile or longer range over land).
Problem for the British is that, as mentioned in other threads, is that the British didn't believe that the 1000lb bomb was needed until 1939/40. They only built 159 of them in 1940 and that was the low HE GP bomb (about 33% HE).
some early British bomb dimensions (rounded off)
Bomb....................length................diameter
1000lb....................220cm..................41cm
500lb......................175cm..................33cm
250lb......................138cm..................26cm
And we start to see some of the problem in designing bomb bays. You can't even fit two 250lbs in the same width as a single 1000lb bomb. You don't what bombs hitting each other on the way out and you need room for a hand to around the side of the bomb to check on the attachment lugs and safety wires (or you need to be able to reach through holes in the bomb bay roof)
Germans don't quite have this problem although the 250kg bomb was a lot more common than the 500kg bomb earlier in the war.
Design the wing with Fowler flaps.
Fowler flaps are really good for getting you into the airfield, while somewhat used for getting out they are a lot less useful
I don't think you want that kind of flap deflection (air brake) when trying to take-off (get up to flying speed). extreme example but Fowler flaps are not a substitute for engine power for take-off.
Airfield restrictions for 1937-39 are not the same when Battle was 1st mooted.
No they are not but they didn't improve a whole lot in the late 30s. That whole scheme about catapulting Manchester bombers? British did a lot of work in the late 30w but then needed more and they were surprised when they went to France and saw what the French fields they were given were like (If they had been good field the French would have kept them)
I'd be okay with 330 mph for the similar Merlin-, 601- or 211-powered design. It will be faster than the bombed-up Bf 109, let alone for the bombed-up Bf 110, that seem to be fared the best among the LW bomb-luggers during the BoB.
And we still have not figured out the attack method, that is to say aiming the bombs and just flying over the area and dropping the bombs somewhere is a several kilometer radius of a local church steeple and offering prayers to god for a hit

You don't have to dive at over 60 degrees and use dive brakes but you need something and something should be better than the what the British did. Their bombsight on the Battles and Blenheims didn't have any stetting for less than 3000ft so the bomb aimer was useless and it was up to the pilots judgement as to when the drop the bombs but they rarely practiced at low altitude and using the pilots judgement.
The whole thing needs to be, in modern language, integrated. It is not enough to fly fast and carry bombs. You need the right bombs for the right targets and you need a way to delivery them with a fair degree of success and you need to practice it.