Don't know about best, but what I'd try is:
Rebuild to "tapeset1", starting with the first tape, expecting it to fail at the end of the tape as you've described
Rebuild to "tapeset2", starting with the first tape, expecting it to fail...
Take tape 1 out of the drive!
Repair "tapeset2" and, when it says insert tape 1, mark tape 1 as missing and continue with tapes 2 and 3
You've now got the most data back that you can, albeit in two sets. You could then try and combine them by copying "tapeset1" to a new "diskset1", then "tapeset2" to that "diskset1" -- "diskset1" can then be moved to your newer machine for the conversion.
Your choice as to whether "diskset1" is a File Set or Removable Media Set. You could, perhaps should, copy the two tape sets to two disk sets, convert them to the new format, then combine them -- it'll take longer but might be "safer".
I'd also start to wonder about the chances of me ever needing to go back to data from 10+ years ago and whether it's worth all this extra work! That'd be a struggle between my innate laziness and my OCD, but you may have compliance issues to satisfy.