There are plenty of possible ways to cut down on dice rolling:
1) roll a single hit location BEFORE you roll for which weapons hit: Yes, 'Mechs move around, but say that it locks onto that one spot and deals damage to the area. Not only would this speed up gameplay by cutting dice rolling, but it would speed up gameplay by making things DIE FASTER.
2) Battery fire. Say that each weapon in medium, short, or long shares a single to-hit roll and if you hit with one weapon, you hit with them all. So if you have 3 medium lasers and an AC-20 in medium range, and two ER Large Lasers and an LRM-20 in short range, you roll for the MLs and AC-20 in one roll, and the ERLLs and LRM in another. Perhaps have minimum ranges subtracted from your to-hit for that weapon only, so it might hit with the ERLLs and miss with the LRM if the roll was close enough to a miss. Once again, speed up gameplay by reducing rolls and making things DIE FASTER.
Another benefit: Speed might actually become ARMOR under such a system. If you're making six to-hit rolls with weapons that need 10+ to hit, the odds are going to eventually mean you hit with one or two of them. If you roll once and miss with them all...
Yes, both of those are Battletech heresy. But remember what happened to the last set of people in-universe who worshipped an ancient, dead idea to the extreme? :D