You may well have looked this site up already, if you have, I apologise for doubling up information.
This site:
http://perso.numericable.fr/~yvesjbel/polk_org2.html
Has a general breakdown of the 5th as it existed beetween 1984 and 1992, they have the 7th Engineer batallion as:
7th ENGINEER BATTALION
Headquarters and Headquarters Company, 7th Engineers
"A" Combat Engineer Company (5 Officers, 154 EM)
- Company Headquarters
- 3 Engineer Platoons (M113A1)
"B" Combat Engineer Company
- (idem)
"C" Combat Engineer Company
- (idem)
"D" Combat Engineer Company
- (idem)
"E" Bridge Company (5 Officers, 146 EM)
- Company headquarters
- Armored Vehicle Launched Bridge Platoon
- 2 M4T6 Bridge Platoons (or Mobile Assault Bridge)
Given the very brief TOE, there's plenty of room to have a unit like the one you describe, especially given the attrition that the batallion will have suffered.
In the British Army all enlisted members of the Royal Corps of Engineers are called Sappers. Their training appears to be as Combat Engineers first and then they are given a trade that seems to reflect the civilian construction industry. In addition, there are Commando Engineers that support the Royal Marines and Parachute Engineers that (you may have guessed) support the Paratroopers, these seem to specialise in combat engineering under fire.
Just to complicate things, the British army also have units of Pioneers, these troops are also trained in light engineering and construction as well as preparing combat positions. I'm not sure what roles they had initially, but from my reading of information and the fact that they are in the Royal Logistics Corps, they now tend to handle less combat oriented roles than they used to.
The Pioneers appear to pride themselves on being equal parts fighter and engineer so this may not be the case.
I like the concept you have for your gaming party, I'd suggest not worrying too much about fitting them perfectly into an ORBAT, it's not stretching the realms of credibility to suggest that a commander gathered a group of tough and skilled individuals together in order to get the job done when equipment and supplies started to dry up.
Good luck in your endeavors.