So far, this only established (by example) an upper bound for the final answer, and the only proof of lower bound is "I tried lots of ideas to lower it, and none of them worked". Lacking better ideas for proof of impossibility to establish a lower bound, I converted the calculations in the spreadsheet to an exhaustive computer search. This found
LOTS of solutions for a 260 limit (to confirm the program was working), which I aborted - sample output:
...
1 x 260, 2 x 30, 3 x 21, 4 x 8, 5 x 7, 6 x 4, 7 x 4, 8 x 2, 9 x 2, 10 x 2, 11 x 1, 12 x 1, 13 x 2, 15 x 1, 16 x 1, 18 x 1, 23 x 1, 25 x 1, 39 x 1, 40 x 1, 59 x 1, 160 x 1
1 x 260, 2 x 30, 3 x 21, 4 x 8, 5 x 7, 6 x 4, 7 x 4, 8 x 2, 9 x 2, 10 x 2, 11 x 1, 12 x 1, 13 x 2, 15 x 1, 16 x 1, 18 x 1, 23 x 1, 25 x 1, 39 x 1, 41 x 1, 58 x 1, 160 x 1
1 x 260, 2 x 30, 3 x 21, 4 x 8, 5 x 7, 6 x 4, 7 x 4, 8 x 2, 9 x 2, 10 x 2, 11 x 1, 12 x 1, 13 x 2, 15 x 1, 16 x 1, 18 x 1, 23 x 1, 25 x 1, 40 x 2, 58 x 1, 160 x 1
1 x 260, 2 x 30, 3 x 21, 4 x 8, 5 x 7, 6 x 4, 7 x 4, 8 x 2, 9 x 2, 10 x 2, 11 x 1, 12 x 1, 13 x 2, 15 x 1, 16 x 1, 18 x 1, 23 x 1, 26 x 1, 30 x 1, 48 x 1, 59 x 1, 160 x 1
1 x 260, 2 x 30, 3 x 21, 4 x 8, 5 x 7, 6 x 4, 7 x 4, 8 x 2, 9 x 2, 10 x 2, 11 x 1, 12 x 1, 13 x 2, 15 x 1, 16 x 1, 18 x 1, 23 x 1, 26 x 1, 30 x 1, 49 x 1, 58 x 1, 160 x 1
...
Many more solutions for a 259 limit, so as I went to post an update with this, I ran it for a 258 limit. This initially found a single solution, so speculating this might be a unique solution, I started writing the current update, and just as I was writing it, another batch of solutions were found... full output at time of writing:
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 2, 17 x 2, 21 x 1, 23 x 1, 27 x 1, 34 x 1, 41 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 15 x 1, 16 x 1, 17 x 1, 21 x 1, 23 x 1, 27 x 1, 34 x 1, 41 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 15 x 1, 16 x 1, 17 x 1, 21 x 1, 23 x 1, 27 x 1, 35 x 1, 40 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 15 x 1, 16 x 1, 17 x 1, 21 x 1, 23 x 1, 28 x 1, 34 x 1, 40 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 15 x 1, 16 x 1, 17 x 1, 21 x 1, 24 x 1, 27 x 1, 34 x 1, 40 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 15 x 1, 16 x 1, 17 x 1, 22 x 1, 23 x 1, 27 x 1, 34 x 1, 40 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 15 x 1, 16 x 1, 18 x 1, 21 x 1, 23 x 1, 27 x 1, 34 x 1, 40 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 15 x 1, 17 x 2, 21 x 1, 23 x 1, 27 x 1, 34 x 1, 40 x 1, 57 x 1, 158 x 1
1 x 258, 2 x 29, 3 x 22, 4 x 7, 5 x 6, 6 x 5, 7 x 3, 8 x 3, 9 x 2, 10 x 1, 11 x 2, 12 x 1, 14 x 1, 16 x 2, 17 x 1, 21 x 1, 23 x 1, 27 x 1, 34 x 1, 40 x 1, 57 x 1, 158 x 1
Trying [...]
The code was sufficiently poorly optimised (and/or the search space sufficiently large) that it didn't make much further progress even when left running for a couple of days. I found additional solutions when running searches starting from different numbers of size 1 bags, including some with fewer bags (349 or 348 bags total rather than the 350 bag solution at the top of this post). I think it would take many years to complete the search with that code.
I also ran searches for 257 and 256 in parallel and got no results. @RobPratt used a different technique that appears to prove 258 optimal.
I personally find it mildly disappointing that there wasn't a unique optimal solution - there seem to be at least a couple of dozen...