So this would emphasize the importance of Insurgent bridges, leading to defense units guarding them. Another aspect of teamwork is lovely.
However, if supply crates were used, the Insurgents could still use the engy kit, using the old crates from a previous British passing. This emphasizes the BAF protecting the British engineer. Maybe even disposing of any traces. Also, like you said, certain Insurgent vehicles could carry crates.
I say the non-lethal technicals on Basrah start off with no crate, and they must arm it off a cache, simulating stowing supplies in the back of the pickup. Once packed up and ready to go, the technical can go to a bridge, drop the crate, resupply, return. Then the Insurgent engineer would set to work, relying on his team to protect him.
There can always be bright ones in a box full of dim bulbs. A rare Insurgent or two with the engy kit could know how to repair a bridge. It's not as far fetched as every single Insurgent knowing how to fix a bridge.
I'm seeing the negatives. A beginner could set off a bomb car over the bridge or a tard could purposely destroy them improperly to hinder the team. Those inevitabilities aside, this new take on bridges would be interesting.