I guess a big question is, what is the definition of an Infantry Support 'mech? I'd say something that keeps speed with the infantry but can respond to a threat on the board. Infantry damage does a hell of a lot to other infantry, and is great for critseeking against opponents, but isn't any good for holepunching. Infantry ranges tend to be pretty short, as well, outside of rare specially-made custom platoons. Their mobility is slow, but they can enter pretty much any hex and don't have facing unless they're carrying field guns.
Speed: 4/6 is about all you need, maybe 5/8 for the faster BA or mech/motor units. 3/5 might be enough for a foot infantry unit, though jump jets would be good in that regard.
Armor: Lots of armor is good. What's threatening your infantry can come in a lot of packages, though again outside of a few specialists the biggest threat to infantry is other infantry. A Bug mech supporting a platoon or two is going to find itself in trouble against a single tank, even if that tank doesn't have any anti-infantry capability.
Mobility: Infantry can get through obstacles like buildings much easier than 'Mechs, so something on the 'demolition gun' side of things to clear paths can help. Jump jets, also, will let you keep up with your troops if they suddenly have to scuttle two streets over.
Range: Here's a tough spot. Most infantry platoons don't outrange the vehicle anti-infantry weapons used on them. Take your standard rifle platoon, your maximum range is 3 hexes just like the classic MG and flamer. Any attempt to get into range on a target means you're facing the guns. Any attempt to eliminate your infantry, though, means the enemy has to get close as well. This invites a lot of short-range fire from larger weapons, since most threats to your infantry will be nearby. Long range guns might not be as necessary on such a support unit, if it can get close.
Special abilities: Most infantry can't start fires or use smoke. Something with an SRM barrage can provide this support; being able to stick a column of smoke between your troops and a Piranha that just showed up can mean the difference.
So we're left with a moderate speed, high armor, very large damage weapon carrier. Obviously you're going to want some anti-infantry capability yourself, with decent range, as well as those special munitions missiles bring. SRM-6s make great smoke clouds, and LRMs bring minelaying options to assist dug-in infantry in area denial against mobile threats. Mech Mortars are useful as well. A heavy gun like an AC20 of any flavor to punch holes in armor to let the infantry close and exploit is on the table, though a RAC/5 or HAG30 does a fair job in critseeking and holepunching.
I guess more or less anything that's a big-gun, mid-speed heavy or assault with a good mix of weapons but a focus on holepunching. Gauss, cERPPC, whatever/20 autocannon, and a mixed spread of secondaries for utility in its support mission.
To be fair, this starts to feel more like it's the infantry supporting the 'Mech, bringing some extra critseeking power and a small denial bubble.