Frame challenge regarding the number system:
It is possible that base-36 is prohibitively impractical in use. They would rather resort to base-6.
The problem with base-36 (and all base-n, where n is large) is that it is hard to learn and use reliably. To be able to do addition, you need to learn all combinations of n + m, where m and n are any of the digits. m = 0 is trivial, and m = 1 comes from learning the sequence so that is easy learn as well.
In base-10, there is surprisingly little to learn compared to base-36. You need to learn in what order the 10 digits are (piece of cake, preschoolers can learn it), and then 2 + 2, 2 + 3, 2 + 4, ... 2 + 9, 3 + 2, 3 + 3, 3 + 4, ... 3 + 9, ..., ..., 9 + 2, 9 + 3, ..., 9 + 9. That makes 8 * 8 sums that you need to be able to come up without thinking, correctly, otherwise you are clumsy and error-prone.
Half of it can be ditched, though, because m + n = n + m, so in order to be able to basic additions fluidly, you need to learn 32 (8 * 8 / 2) "primordial additions" that all additions are based on, and you need to learn them absolutely correctly. You cannot make such mistake that 3 + 6 = 8.
In base-36, it becomes much less manageable: 2 + 2, 2 + 3, ..., 2 + 33, 2 + 35, ..., ..., 35 + 35. After ditching half of this, we still have 578 (34 * 34 / 2) distinct nontrivial additions between mere two single digits, and you need to learn all of those, otherwise you are pretty much screwed when doing anything with bigger than single digit numbers. You cannot make such mistake that D + J = X. It is W.
That makes base-36 very impractical if you cannot write your numbers. If you have to adopt them form another culture, then you have been in serious trouble with your base-36 until you got the numbers.
The aliens would be better off with base-6. This has many advantages: there is now only 8 (4 * 4 / 2) single-digit additions that need to be learned, and then you can do addition. There is little to memorize, and it is easy to transmit also orally without any written numbers.
In addition, if your aliens do have 36 fingers, they can use them easily in groups of six, giving them ability to display 6 distinct numbers just with the 36 fingers, ie. it goes up to ZZZZZZ (if you borrow the English alphabet). That is, they can go up to 2176782335 (instead of 36) using this system that is also easier to learn and less error prone.
If they had two subgroups, one using base-36 and other base-6, the base-6 group would have a significant advantage over the other, and base-36 would be dropped eventually in favor of base-6.