The A-20 itself seems like it might have fared well enough in a mosquito-like role, though the internal bombload was more limited. (more limited than the Ju-88's maximum internal load too, though not limited to the small, vertically oriented bombs as on the Ju-88 )
The DB-7 flew for the first time about 15 months before the Mosquito (and about 10 months after the DB-7B, which was just about a totally different aircraft) And the DB-7s first used 1000-1100hp for take off R-1830 engines with single speed superchargers. They were good for 900hp at 12,000ft. The DB-7/A-20 bomb bay was larger than the Mosquito's and somewhat more flexible. Later A-20s could carry pretty much the same "standard" bomb load as a Mosquito, four 500lb bombs.
The later A-20s got a 140 US gallon fuel tank in the upper part of the bomb bay which did NOT affect it's ability to carry the four 500lb bombs inside. Mosquito still carried more fuel. Even later A-20Gs could take 325 US gallons in the bomb bay and still hold the four 500lb bombs. JU-88 had a choice, small bombs inside (carried horizontally not vertically) and about the same fuel as an A-20 without upper bomb bay tanks or cut way down of the internal bombs and carry more fuel.
Switching from R-2600s to R-2800s would be less dramatic, though more so if consider 2-stage or turbocharged R-2800s. The dimensions and weights of the A-20 are fairly close to the F7F ... if obviously not as clean of an aircraft.
Well, having an internal bomb bay will do that, make for an aircraft that is not as clean. So will using older cowling designs and older airfoils and..............
Still, it might have already been close enough to be modified (without a total redesign) into an unarmmed fast-bomber configuration with more powerful engines. (logistics of turbocharger supplies or 2-stage R-2800s would have limited matters, though, and just stripping down the existing A-20s probably wouldn't have gained enough performance to be really good 'fast' bombers)
They broke a B-25 re-engined with R-2800s by pulling up too sharply at the end of a low altitude high speed pass.
A DB-7 went about 15,030lbs loaded (Normal, not overload) or less than the F7F did empty.
Now please remember that US light/medium bombers were commonly built to an ultimate stress factor of 6 "G"s. Or 4 "G"s in service with a 50% safety factor.
A-20s with R-2600 engines went to about 20,000lb pretty quickly and had the wing structure reinforced to handle the higher weight (they also got the larger tail)
Later Versions went to 24,000lbs with a 27,000lb over load rating. Many bombers were restricted in the maneuvers they could perform to begin with and in over-load condition even more restrictions are mentioned in pilot's manuals.
How much beefing up the later ones got I don't know.
You can certainly beef up the structure to handle the R-2800s but at what cost in weight? and engineering time?
The engines used in the majority of A-20s went about 1940lbs.
R-2800s as used in early B-26s went about 2270lbs and the two stage Navy engines went about 2480lbs
without intercoolers and ducts. Of course you need bigger props, engine mounts, exhaust systems etc. Power plant is going to go up a lot more than dry engine engine weight and that is already 1/2 ton.
Leaving out seven .30 cal Brownings (at under 25lbs each) and their ammo ( 65lb per 1000rounds?) isn't going to cut it.
I don't think I am smarter than Edward Heinemann and if he thought in Jan of 1941 that starting over with clean sheet of paper to start designing the A-26 was the way to go I am certainly not going to say he was wrong. The Xa-26 prototype went 21,150 pounds empty. How much of that was due to the gun turrets I don't know but "empty" weight often includes the turrets but not the guns.