Success!
Well, it's been almost 2 years since I started this project, but it's essentially done now. It only took a few hours to do the actual work and find parts.
Here's what it looks like now:

For anyone interested, here's a summary of what I did and some pictures that may help others working on similar pre-war Lionel Multivolt transformers.
Lionel Type T Multivolt Transformer Repair Notes:
Due to the possibility of asbestos, while wearing a respirator, I clipped off the original line cord and discarded it along with the brittle Bakelite wall plug.

Next, I tested the primary and secondary coils for continuity. So far so good.
Bought a new 18-2 stranded line cord with cloth cover (woven over PVC jacket) and a modern vintage looking non-polarized plug. Cleaned the top panel wiper contacts & porcelain insulator, then after stripping and tinning the wire ends installed the new cord and plug. Note the strain relief knot, 12 gauge solid insulated wire wrap and heat shrink tubing.

Tested the insulation on the windings without any loads connected to the secondaries. Using a Variac, slowly brought the input voltage up to 130V AC input to ensure no shorting. I kept an eye on the ammeter to ensure no current was flowing during this test.

Tightened Output binding post retaining nuts on the top panel.
On the inside of the top panel, the variable (wiper) contact post internal nuts were especially loose and in some cases not making good electrical connections.

At 120VAC input, tested all secondary fixed and variable outputs and verified correct the correct voltages were present.
Using resistive Loads, I tested the outputs one at a time up to 80 Watts each for about a minute, with no noticeable heat or unexpected sounds coming from the transformer housing. Under load the measured voltages dropped a bit as expected.
Since it wasn't obvious at first, while reassembling I needed to use the original dis-assembly pictures I took, to find the correct locations within the lamination stack where the core support brackets were supposed to be so that everything had the correct clearances inside the housing.

For anyone doing this, make sure that the 2 horizontal pieces of cardboard are put back in between the backs of the terminal screws and the secondary leads to help protect the shellacked cloth insulation on these wires.

The next steps in the assembly were to reinsert the fragile cardboard that wraps around the four sides of the core and secondary wires and carefully fit the assembly back into the housing.
Before bending the 4 retaining tabs back onto the top panel (not wanting to stress them with multiple bendings), I temporarily zip tied it in place and did one more electrical test of everything to make sure there were no issues caused by the reassembly process. One verified, I bent the tabs back down and removed the zip tie.
Now it's essentially done except for some possible cosmetic finishes.
It's still missing one of the Binding Post thumb nuts. Although they're not an exact match, I'm probably going to get some Repro-Nickel replacements from Trainz. Link here: https://www.trainz.com/products/lionel-1033-47-6-32-fancy-nickel-binding-post-nuts-fits-1032-1033-lw?variant=43592738013306
Maybe at some point I may try touching up the white lettering, but my fine detail painting skills aren't great and I don't want the lettering to look blotchy. Besides, other than being difficult to read, I kinda like the well used look it has. If you have any lettering touch-up tips to share or have an original prewar thumb nut to spare, that would be great!
When I use this, it will only be occasional and while closely supervised. For added safety, I intend to plug it into a fused 1Amp supply circuit.
I want to again thank all those who contributed their knowledge and experience here and hope this Topic may be of some help in the future to others.