No your memory serves you right and you are correct. The 3rd did have their permanent engineer unit, the 10th Engineers, then there were also attached independent engineer groups such as the 540th and the 36th CE's.
Just to give you an idea, these are the units that the 540th were in direct support of, or attached to during the ETO. Please note that your 7th Inf Reg is listed below.
9th Infantry Division - 47th & 60th Infantry Regiments
1st Division - 18th Infantry Regiment
2nd Armored Division
82nd Airborne
Darby's Rangers
3rd Infantry Division - 7th & 30th Infantry Regiments
36th Infantry Division - 141st, 142nd, & 143rd Infantry Regiments
British 5th Division
14th Armored Division - 62nd Armored Infantry Battalion & 125th Armored Engineer Battalion
117th Cavalry
45th Infantry Division - 179th Infantry Regiment
Proud Daughter of Walter (Monday) Poniedzialek
540th Engineer Combat Regiment, 2833rd Bn, H&S Co, 4th Platoon
There's "No Bridge Too Far"