These kinds of positions are very tricky.
If White had a two-roll position then 18/10 would be clear since it would allow you to bear off with both 66 and 55 rather than just 66. However, since White has a three-roll position, immediate doublets are crucial only if White rolls doublets. Here you'll usually have two rolls to save the gammon, so you need to consider that scenario first before worrying about which doublets get you off immediately.
A trick that I learned from a Gammon Village article by Doug Zare is that it is sometimes helpful to mentally give yourself enough extra 6's to move all your checkers into your outer board, and then for each such 6, add a checker on your 12pt. Finally, take a checker from your lowest occupied point in your home board and move it 6 pips backwards. Then look at the configuration in your outer board and see what it looks like as a bear-off position. Applying this trick to 18/13 15/12 gives us this:
| White | 0 |
|---|---|
| |
| Blue | 38 |
And applying the trick to 18/10 gives this:
| White | 0 |
|---|---|
| |
| Blue | 38 |
This trick isn't perfect but it is often suggestive. Here we see that it suggests that 18/10 should give you better diversification. My guess is that if White had a four-roll position then again 18/10 would be the better play (though only by a little bit since you'd be a strong favorite to get off the gammon regardless).
However, White doesn't have either a two-roll position or a four-roll position; she has a three-roll position. Thus to calculate the better play, one has to actually count pairs of sequences. In particular, unless you roll doublets, you need to get a crossover for every die you roll. If you play 18/10 then any 2 or 1 will mean you'll lose one of your crossovers. This fact should tip the balance in favor of 18/13 15/12. That is, it's so important here to get crossovers that it's worth putting a checker on the 13pt even though that's normally a "wasteful" location for it.
When you roll this out, I'd suggest changing the roll to 22 so that putting the checkers on the 14pt and 11pt is also an option.