A lot of parts are still available for your bike. You've got a dual pin crank. If you look at the parts breakdown and click on a part and scroll down on the right side it will tell you what bikes that part is used on. It sounds like the shop doesn't what to take the time to diagnose what is actually wrong with the bike. Not sure what the difference between the t and the c would be but it couldn't be much. Hopefully someone will come along with a bit more experience with this like @CSRoad or @cyclejohn

https://www.partzilla.com/catalog/honda/motorcycle/2001/vt1100t-a-shadow-1100-nl-american-classic-edition-tourer

 

Don't add me in there it's an 1100, I nominate @oldguy for possible replacement engine guidance.
I would put in a good salvage engine. I think money wise it is not worth building an engine for a 22 year old bike if you are paying to have it done. It may be justifiable with a swap and a not too expensive hourly rate.

 

The 1100s have hydraulic lifters. So no valve adjustment needed normally.
I am suspicious of that low compression unless the engine has been badly overheated.
But with only 6 percent leak down on the rear that is excellent.
15 percent is a bit low but not so bad as not to run fairly well.
So I would go back and test compression again.
Wide open throttle .
Then do it again with a small amount of oil in the cylinder to see how well it reads then.
Make sure you are taking only one spark plug out of each cylinder.
And see what the spark plugs look like.
The repair shop said burning oil so you may find fouled plugs from that.

 

what years of this model (or others) have same engine.

New shop says what it would cost to modify a C engine to T is not worth it

Any VT1100 dual crankpin engine from 1998 to 2007 will fit in your Ace Tourer (Spirit, Sabre, Ace Tourer).

I have put a Sabre engine in an Ace Tourer and I have put an Ace Tourer engine in a Sabre. The only difference being one of the crankcase halves (lower, rearward, right side) bolts is a hexagonal male/female stud also that provides a mounting point for the exhaust hanger, so, you just have to move that one stud from one bike to the other and that is it. No other differences.

I know 2 people that have put Spirit engines in their Ace Tourers. They put Spirit engines in their Ace Tourers because the final driven gear ratio allows for reduced RPMs at all speeds and that really shines on longer rides where the bike is in top gear most of the time (lowers RPMs by 300 at those speeds) and feels less buzzy. I have toyed with the idea of purchasing a Spirit engine to put in my Ace Tourer but most of my riding is in the foothills and mountains of NC and Va. and the gear ratio for that is better with the T engine.

The Spirit engines are painted silver and the other VT1100 engines are painted black with just the outside of the fins showing silver. If that matters to you at all?

I don't know if a single pin VT1100 engine is a direct fit so I won't comment on that. I have no actual experience doing that swap, just the other ones.

Anyway, hope this helps.

 

The Spirit models have a taller secondary gearing as cycle john mentioned. Replacing the the 170/80 tire with 180/70 will alleviate that issue.

 

The single pin engines would also need the single pin ignition module and wiring, more trouble than it's worth, plus they are usually less power. 1100 Spirit or Sabre engines will fit and work, as will the later 1100 Aero (2001 +?) you can use a lower profile rear tire to help with the Spirit's gearing. Or taller tire for the Sabre's gearing. Won't be exactly like you had, but works fine.