IMO, Zook and PIATs are fine as-is.
2x Bar could work with a long-range nerf and buffs to short-mid so its closer to the STG or Con PPSH. It would also promote synergy with 1919 Rifles or Paras ideally.
1919 and Bren limited to one.
I've had a friend argue that K-Guns should remain as-is to prevent Special Weapons from being a doctrine everyone has and no-one uses. It'd be interesting to test, but I wouldn't be surprised if that winds up being a bad idea.
I agree with you in general but the Bren and Vickers K need some better balancing in addition to what you've proposed.
The Bren has three variants (Tommy, Sapper, and Commando). Unfortunately, picking up a dropped Bren always gives you a Tommy Bren regardless of the squad. In addition, the Tommy Bren is the weakest variant having roughly 87% of the DPS of a Sapper Bren (a Sapper Bren is roughly equal to an LMG42). The Vickers K is identical to the Tommy Bren. Please note that the Tommy Bren's DPS stats are quoted as an in-cover value.
I'd suggest:
- Fixing the pick-up issues
- Giving Infantry Sections the Sapper Bren
- Giving Royal Engineers a Bren with an assault rifle weapon profile (including firing on the move)
- Giving the Tommy Bren a slot size of 2
- Mirroring the Vickers K with the new Royal Engineer Bren
This would provide REs with a weapon upgrade that fits better with their initial weapons and give Infantry Sections a weapon worth its cost.