Not sure they have unreasonable delta-v though, given that space combat in BattleTech seems to happen in relatively low velocities in general.
We're about to
science 7th grade math the shit out of this.
Considering the case of an on-board engagement (not going to get into high-speed closing engagements here because the math is way more complex and it requires even more crazy-high delta-v on the part of the missiles), the range of a Barracuda is 50 hexes. A space hex is 18,000m and a space turn is 60 seconds. Putting those together, we find the
minimum average speed of the missile is 15,000 m/s. It's actually higher than that because the missile almost certainly doesn't have the full 60 seconds to reach its target, but we only need the minimum possible speed for the sake of argument.
We're going to ignore the velocity of the mothership because the rules don't adjust weapon ranges based on speed in on-board engagements, and a ship with velocity 0 can still engage any target within the 50-hex range bubble.
The average velocity of an object accelerating from a standstill is half the final velocity, so the required final velocity is 30,000 m/s. That gives a minimum acceleration of just under 51g, which is an entirely reasonable acceleration value for a missile. The upper bound of acceleration for a missile that won't crush itself like an empty beer can is probably around 150g. By increasing the acceleration you can hit top speed sooner and then coast, which does reduce the final velocity requirement. With 150g and a boost-and-coast profile you could get the final velocity as low as 18,000 m/s (12 seconds spent accelerating followed by 44 seconds of coasting).
So, depending on the acceleration our rocket motor can provide, we're looking at a minimum delta-v of between 18 and 30 kilometers per second (that's 2-3 times the total delta-v budget required to get from the ground to low Earth orbit).
Knowing this, we can plug the information into the rocket equation, dV = Ve * ln(launch mass/final mass), and solve for various propulsion methods.
Exhaust velocity for a chemical rocket is in the range of 2500 m/s, for solid-fuel rockets, to 5000 m/s for the most efficient liquid-fuel combination ever studied. To reach 18 km/s the liquid-fuel rocket would need a mass ratio of ~36.5, that is to say it would be 97.2% fuel. At the other end, for a solid-fuel rocket to reach 30 km/s it would require a mass ratio of 163,000 (!). That's a missile which is 99.9993% fuel!
Fission rockets have exhaust velocities as high as 10,000 m/s, which gives a mass ratio between 6.1 and 20.1 (83.6% to 95% fuel). A gas-core fission rocket could get the mass ratio as low as 2.46 (59.3% fuel) for the lower-bound velocity, but the radioactive mess created would make them problematic to use anywhere near a settled planet.
Metallic hydrogen would have similar performance to a fission rocket, if it's even possible--but it would still require tons and tons of the (expensive) stuff per missile, and the properties that make it a good candidate fuel also make it a good explosive and it's theorized to detonate spontaneously due to quantum effects, so making it stable enough to use as a fuel without the whole missile exploding in the magazine would be no mean feat. And if they had access to something that energetic, they wouldn't just be using it as space missile fuel.
BattleTech's magic fusion engines have dramatically higher exhaust velocities than any of the above technologies. One ton of fuel provides a 30-ton fighter with 40 g-minutes of delta-v, that is about 23,500 m/s. An effective exhaust velocity of ~695,000 m/s! Even if capital missiles are using fusion engines just 10% as efficient as the ones in aerospace fighters, that's still 3x the performance of the best fission rockets and 20x better than typical chemical rockets.
The BT fusion engine is also efficient enough to raise the missile's velocity to the point where it can hit a max-range target in just a few seconds rather than a full minute. None of the other technologies make that practical. Even gas-core fission engines couldn't deliver the performance to hit a target at 900km in under 30 seconds.