Don't see how the 501 would be useful for the Demo, though if we're to talk about weapons already in the game, here's my personal list of weapons that should be cross-perk
Singular 1911 Pistol for Commando: I still find the usefulness of the 1911 questionable since the 1858 buff from the summer of 2018. Giving a singular 1911 to the Commando would at least allow a 12 round pistol with 77.5 damage per round (Assuming that the player is Level 25 and is using both Hollow Point Rounds and High Capacity Magazines)
Hemogoblin for Demolitionist: I still think the most prudent move would be to rework the damn thing to function as an explosive weapon with heal darts. Usage of the upgrade system could potentially allow an explosive-based Hemogoblin to act as a sidegrade to the RPG-7.
Seal Squeal for Sharpshooter: Using this thing honestly feels more like a Ballistic weapon first and an explosive weapon second. I reckon that a Sharpshooter would want to use the harpoons to build up a Rack 'em Up stack, while also activating Sniper and Stability prior to the detonation of the harpoon explosives. Plus, it would really be nice to see an end to the Sharp vs Demo debates and I think the most efficient way to do this is to have a Sharp/Demo cross-perk.
Healthrower for Firebug: As a weapon, the Healthrower actually sucks. I feel like this weapon exists people seem to have this uncanny ability to dodge medic darts despite the auto-aim mechanic tied to them. I'd like to think that if it were to benefit from the Firebug's perk skills, the Healthrower would have greater killing potential, while also allowing the Firebug to carry a Medic Weapon alongside the Helios.
Freezethrower for Firebug: See
this thread in the Steam forums
As for new weapons:
Revolving Rifle for Field Medic/Sharpshooter/Gunslinger: I dunno why, but i feel like a DMR for the Medic could actually work as a high damage tier 2 in the form of a modified revolving rifle chambered to fire syringes (The downside would be that it uses ammo for healing, rather than regenerative darts). Also, the Gunslinger totally needs a third rifle.
Glock-18 for SWAT/Gunslinger (pistol only)/Berserker (Pistol/Shield): Because people won't shut up about giving the SWAT a riot shield, and i still think it'd just be an SMG that can be used for parrying.