Given my current post and minor obsession on my Arkansas notes and banks that issued sheets of 4x$10s (as opposed to sheets of 3x$10-$20s), I checked on Marietta, OH The Central NB Ch. # 5212 and Philadelphia, PA The Quaker City NB Ch. # 4050 and found both ordered sheets of 3x$10-$20.  So I still do not know why the Back Plate numbers differ and I still don't have a scope with a camera.

Once again, my compliments to your eyes. Ever thought about being a grader at PMG? I would recommend you. I would like to think I would have noticed the T's not being crossed on your Marietta note, but I can't say with 100% certainty. I always assumed that there were multiple back plates for an issue like the $10 NBN? I presume your evidence is proof of that fact? Or perhaps just the number of the back plate was re-etched? I expect you will find n answer soon and educate us. 

Imagine grading the Bebee collection down in Sarasota!  That would have been fun.  I'd be the one stuck grading 10,000 consecutive 2019 $1 notes courtesy signed by Pelosi and Trump... ugh.

One question I have (Q1) is if the BP #s for $10 4x$10 sheets started at 1 to n and if the 3x$10-$20 sheets did the same.  Thus my check above on the Philly and Marietta bank's choice--both ordered only the 3x$10-$20 format, so no answer there.

Then the question (Q2) is how much variation is there on a sheet (limited by the number of sheets to examine).  

Your New York, NY The Hanover NB $10 RS is interesting since the Hanover issued both formats.  I can't quite make out the bp number (why doesn't PMG record BPs?).  I think it's a 6 or 9; definitely 1 digit.  So, that's likely from 1906 or 1907 as the 4x$10 format was first produced in 1906.  So the task is to search archives for other $10 RS from The Hanover and find the same 6 or 9 bp.  Based on the treasury SN, you can tell what format the note came from.  And we might learn Q1 above.

I probably should spend my time looking for engraving errors like the pcblic error on Woodchopper notes :-(  That's a tough one to see.

The bp on the Hanover note is 9 if that helps. Is there any documentation anywhere re: the uncrossed t's on the Marietta notes? Or did you pick that out? Good point re: PMG. That's my luck as well, to grade 10,000 $1 notes  I have one or two graded $1 FRN's, I got one because it's the only 68 in my collection. Frankly, I don't see the point of amassing every recent Fr. FRN especially of the $1 ilk. I see some of the collections here and the sheer bulk scares me! They must have kilos and pounds of graded notes. I guess it must be fun for some, just not my cup of tea although I do like Mnuchin's signature. Quite unusual compared to most Treasury Secs. I try to avoid looking at the Bebee collection. Where one note is more valuable than my entire collection and there are hundreds more. It is quite phenomenal. I would guess there is easily 100 million dollars if they ever went to market. On another note, I thought there would be some bargains at auction after the pandemic correction but not really. I saw a few at the recent LK auction but I forgot to bid. Probably why there was so many bargains! People just forgot in between all the Zoom conferencing  

Seems more engravers did not cross their ts then did.  Could be the style taught at that time, I don't know.  I got Cs in penmanship.  So I was lazy last time and didn't look at Heritage and the answer is there with only 3 $10 red seals on The Hanover NB (including yours) archived.  All three are from 4x$10 plates.  Here are the SNs plate letters and back plate numbers:

F-VF 5497/A5497 pp F/2
PMG 30 SN 18309/A390378 pp G/9
PMG 25 SN 22418/A396987 pp D/1

Since a note from a 4x$10 plate (first use was in 1906) has a bp number 1, they obviously started at 1 despite having used bp 1 on 3x$10-$20 sheets a few years earlier.  Face Plates D, E, F, and G were made (only missing an example of the E in the archive)  The odd thing is bp 1 showing up on SN 396987!  They got a lot of use out of that back plate!!   The best note is the one with the bank SN = treasury SN which proves The Hanover was the first bank to order sheets of 4x$10s [I'm surprised Heritage did not note that].  That's quite a large order.  5497 X 4 X $10 = $219,880 (at least).  Maybe you could check T&P and see if there are any more $10s there (only T&P doesn't record treasury SNs  ).  So need another source of data...  oh well.

I guess I should join in on LK auctions.  I never have.

Indeed, there are quite the population from T&P. Ch #1352 reports 15 $10 RS with a total population of 169 notes. There were no small notes printed for this bank. There are 12 $5 RS and no reported $20 RS. The highest SN for the $10 RS is 37750. Two of the $5 RS are SN 1 at pp B and D. It is quite frustrating that T&P doesn't report the treasury SN as it seems it would be as easy as the plate # which is reported. Go figure. I'm not sure about your logic on how to recognize that this bank was the first to order 4X$10's? Definitely check in on LK as sometimes notes are comparatively cheap. I don't know why. I picked up a couple of AU common small notes from Denver 7408 and Easthampton 428 for $175 and $90. 

Pretty sure The Hanover NB was 1st.  I know because of this note in Heritage's archive:  $10 Red Seal, F-VF  SN 5497/A5497 pp F/2.  I know (for sure) it's from a 4x$10 sheet because the font on the serial number is the new font introduced in 1903.  If it was from a 3x10-20 sheet it would be the old font for SN A5497.  Then since the bank SN 5497 = the treasure SN A5497, its from the first run.  I don't think any treasury SN exist without the prefix letter.  A prefix ought to be the first run. 

For 3x10-20 sheets, it wasn't until treasury SN B241777 that the new paging machine was used with the new font sometime in 1903.  It would be good to know if the Hanover only had plate letters A-C for their 3x10-20 sheets and D-G for their 4x$10 sheets--I suspect so, but I could be surprised.