Given that the beam dreadnoughts are given to be larger and slower to begin with, they would be both more vulnerable to anti engine missiles and lower in numbers, meaning presumably they would require less anti engine missiles to disable. I still think it would fundamentally favor the faster people regardless.
e: Ideally you would build your dreadnoughts with the same engine percentage as the enemy, so given equal tech the smaller ships wouldn't really be faster anyhow.
The dreadnoughts would also have more total engine tonnage and thus be less vulnerable to the missiles, and have more damage control rating so they could get the engines up quicker while the enemy was still disabled. Or as noted, if you have multiple ships you can leave the one with engine damage behind.
And yeah, you could design your dreadnought to be faster, but the point is that in the current system for beams faster ship = invulnerable, and I like the idea of a system that is fuzzier than that - where even if you're slower you can get some punches in, even if you're at a disadvantage.
This is why I raised this very problem in my first proposal to this and said that weapon fire controls would not work for a minute or so after the boost is turned off, much like when you do a combat jump through a jump point. You also could be sitting completely still in space for that time before you can start moving again and be rather vulnerable.
This seems like it would make it completely impossible to ever engage a ship with beam weapons if they didn't want to, since you couldn't catch them, for what it's worth. Basically the current situation but making it so that a faster beam ship couldn't even force a missile ship into range once it was out of missiles, since the missile ship could just turn on its tuners if the beam ship was almost in range, and the beam ship wouldn't be able to engage even if it did use its tuners.
While that is a possible use, the opposite is also true. If you have technological equivalent fleets, a beam weapon fleet is very often the one that is built to be faster. This is by design, after all you need to get in range to shoot.
And so, say you have two fleets. A missile fleet, and a beam fleet. Supposing the missile fleet can surpass the beam fleet point defense (because if not, of course the point is moot), an anti-engine missile like you propose would mean that the missile fleet can escape 100% of the times. Just load a wave of anti engine missiles, disable the enemy ships' engines, leave the system.
It's a get out of jail free card.
Mind you, I would be much less against this if beam weapons were not capped at 1.5 million km range... but as it is, if you take speed away from beam warships, they're literally just expensive junk.
It would work completely the opposite in practice. If they launch a wave of anti-engine missiles and take out the engines on one or two ships, you can leave them behind, and still have 100% beam superiority over pure missile ships. Even if they have such massive crushing superiority that they can disable every engine in your fleet, then A) they could probably just blow up your fleet with missiles before you close the range, and B) you'd be able to repair the engines, probably within less than an hour, and resume the pursuit.
Anti-engine missiles are not symmetrical in nature (one of the primary reasons I like them over tuners). Because of repairs and a fleet being able to leave damaged ships behind, they are far more effective at preventing a fleet from escaping than they are at preventing a fleet from pursuing.