I opened up the 
IAST in my browser and went to 
Settings > General > Keyboards and opened up the Text Replacement dialog. It was a bit tedious, but using copy-paste, I set up a system of shortcuts which would 
allow me to type, for instance, .s and have the iPad replace it with ṣ.
There are two important limitations to this approach, if you want to follow it:
- You can't set up 'S and 's for Ś and ś, but instead you must have a 
distinct shortcut like '^s for Ś. This is because text replacement is 
not case-sensitive. If you want to be able to type capital letters, 
you'll need a different shortcut.
- You need to have spaces between the shortcuts. For instance, if I 
want to type "Bhaṭṭa" I can't just type "Bha.t.ta" if my shortcut is .t 
as the iPad only recognizes the shortcut when it is entered as "Bha .t 
.t a." This means that composition is slower.
I'm not sure why Apple has decided not to enable the ABC-Extended 
keyboard on iPads, and so far I haven't encountered a third-party app 
that allows IAST typing on the OS. If anyone has solved this problem, 
let me know!
Best,
Malcolm