I don't know how I missed that 

The battery box is 2.8" square.  The available height is hard for me to pin down but I can now say that this is essentially the same battery (different branding) as the one I have.  Dimensions for the ML 1.3-12 I have are 3.86 inches x 1.74 inches x 2.32 inches.
There's a .20" thick rubber pad on the bottom of the battery box which if removed will help the height issue, but I don't think will resolve it.
Looking in through available space with a handheld light and closing the seat down can only have me guessing on the addl. needed space.  and I'm thinking it's around a 1/2" +/-

Being that I have the original RT1 battery box, it may be prudent for me to see if that is perhaps a bit "deeper" than the RT2 battery box.  Being that its smaller perhaps its depth won't be as affected by the rear fender.  I'll look into this tonight whereas it just occurred to me.  I think the original RT1 battery box may accommodate the length and width.  So, we'll see on the depth.

Another thought I just had is with the long skinny battery maybe I should consider converting the tool box into a battery holder.  Hmmm.

And lastly, there's the bailout of going back to the wiring and further converting the system to "no battery req'd".  But that'll be my last choice.

I've got MBT6N4 in my DT3 and RT2. Dimensions are 97x71x71mm (ie a bit smaller than the 12v one I showed) and they fit fine. 

I see that the RT1 originally took a smaller battery than the 2 & 3, so maybe the frame and seat geometry is working against you? 

Gr8uncleal - I sure have reason to believe you're correct.  The RT2/3 seats are different.
The reason I have an RT2 battery box was so I could accommodate a nice new 6v Motobatt AGM battery I had bought.
I never got to use it because I subsequently decided to install the VAPE ignition system.
So, the installed battery box may in fact be the culprit.
I do know that there's practically no additional space between the rear facing bottom edge of the battery box and the forward curvature of the rear fender.
The original RT1 battery box - being more slender - may have a greater depth being that it won't have interference from the rear fender's curvature at the same point.
This is a very good point you've made.
My "hybrid" fitting of the RT2 battery box didn't take into consideration the the fact that seat pans are different.
I will be focusing my attention on the battery box as soon as I get home later today.  The battery height may not be the issue so much as the battery box is.
 

I’ve often thought of getting a cordless tool battery to work… My Trail 90’s especially… easy in and out and charging simplicity… imagine how simple for winter storage?
Sneezles61

Hey Pete, give me the dimensions and I'll get over to Merrit Island when I can. Just bought two new projects so they have my attention right now but will make time.

For the record to all interested parties, the battery I have and is for a mobility scooter. It's fairly small so I had to add padding around bottom and sides. I'm concerned that with continued long term/time use that the battery wont be able to handle it. Not being too smart on electrical I could just be paranoid and the battery will be just fine.

Sneezles, you might be on to something.
Earlier today, I was speaking with a friend and fellow collector about this and he thought that perhaps the reason I don't seem to be getting clear-cut advice on what will work is that others that have installed the VAPE system (or other 12v system) had later models than I.  Who knows?
I did make a bit of progress in "studying" the situation a bit more.
I went home for lunch and spent some time digging out the original RT1 battery box.
Two immediate observations:
1)  The depths of the boxes (RT2 & RT1) are the same.  NO HELP HERE
2)  The mounting tabs on the RT1 box extend above the top of the box's sides on the RT1 box.  They're flush on the RT2 box!  THIS ADDS EXTRA DEPTH and perhaps all that I'll need.
The battery I have (pretty much identical to the one in the link Gr8uncleal provided) does fit in the box.
Hopefully I'll be able to get the clearance I need with the extra lowering the raised tabs will provide - which to the eye appear to be just about what I had speculated as being needed... 1/2" +/-
IF this works, I'll have a successful fit.  But not necessarily a proper battery for the requirements.
The battery is only 1.3 amps.
I don't know what happens if lighting amperage exceeds battery amperage.  Even if just temporary.
First things first.
I'll prove out that I have (hopefully) a physical fitment tonight and then go from there.
Fingers crossed.

Sneezles.... I've done it with Nicad... In fact on my MG1-T I soldered together RC car sub-C batteries and fit them into a gutted wet cell battery case.

On other bikes with a headlight voltage regulator I've simply used sub-C 7.2v Nicad RC car battery packs... I had a few old ones lying around. And of course I have the chargers. (I used to race RC cars and trucks... primarily Nitro... some electric... was sponsored for a few years.) Seems to work ok for occasional use. I don't charge the pack too much before a ride.

Someone else on here a few years ago used one of those tool battery "adapters" you can get on ebay and Amazon and used a nicad 12v tool battery. Problem is these days most tool batteries are lithium. You don't want to use a lithium tool battery with a crude Yamaha charging system...

Lithium batteries need careful charging and special chargers. Not using a correct charger with lithium will almost certainly result in explosion or fire.

Hopefully Rex's wouldn't list it with that description if it wasn't suitable.

Vis a vis nothing in particular, Gr8auntiepat went on a "Bat Walk" last week and was looking for a torch. The heavy duty, rechargeable one that we got yonks ago wasn't working and so she used her head torch that she uses when she goes running in the winter. 

Anyway, we decided that we ought to take a look at the non working one and, blow me down, it uses the same battery as that in my DT360A! One new battery later and the old girl (the torch, not Gr8auntiepat!  ) works a treat. 

RT1, I'm pretty sure your battery is 1.3 amp-hours. Not amps.

Amp-hours is a rough measure of how long the battery will power things... so that battery will deliver about 1.3 amps for one hour... or 2.6 amps for 30 minutes...

If you have your VAPE powering the headlight with engine on, that little battery should be fine. If you don't, then it will go dead pretty fast with engine off and headlight on. Still should be ok though.

USA has a law that on a motorcycle built after 1971 or so the taillight (only) needs to be able to stay lit for 30 minutes with engine off. Some states might check for that if you need to get inspected where you live.

The other problem with most battery eliminators (besides no power to taillight with engine off) is they SUCK at idle. Horn and signals probably won't work unless you rev up engine. I've heard they make larger $100+ eliminators that work better but I've never tried one.

Well that's good news Mark, I'll quit worrying about the battery both Pete and I have. Also, one of my battery chargers does have a setting for lithium and that's what I have used after the batter/bike sat for about a year...