How do you handle interactive voice response systems

If the ivr listen to the sound like your voice to do xy thing then the best you can do is use external earphones or airpods so their is no cross contamination with the ivr, that's what I do with automated thing like when calling telus.
Alternatively you wanna consider disabling the Auto-select Speaker in Call option under audio in VO settings so it stays with the speaker / mini speaker you select yourself.

I haven't tried it in the latest updates. The best results I've gotten are with a bluetooth keyboard and the earphones/mic. They will have you pushing numbers at one point or another. VO will start yapping all the numbers you've pressed during the phone call every time you press a number and you can't follow what the phone system is saying. You can't be sure you have pressed the correct screen number if you silence VO, so you can use the numbers on the keyboard with VO silenced.
However, updates in the past have broken the ability to use the keyboard with the phone numbers from time to time. Also, there's the whole quick nav on or off thing that, sometimes randomly kicks in and you might be pressing a key and it tries to move around the screen instead of entering a number.
I usually end up trying to get to a live person.
I don't know if BSI is a better solution because I haven't gotten around to trying it yet.

Unless you are really good at BSI, you do not want to use that for having to type numbers in an IVR in instance. Because for every single number, you have to do the #. Of course if you are doing a series of numbers, like a phone number, credit card number, whatever, then you only have to do it once. So there is that, at least.
I would say use the above comments. Both of them. Put in an earbud, and use yourself a portable Bluetooth keyboard if you must. Better yet, disable the quick NAV hot key, so you don't make the mistake of accidentally activating quick NAV while trying to use an IVR.

HTH.

For BSI, couldn't you just use computer braille? This gives you direct access to the numbers by transposing them down a dot. So for instance, instead of the number 2 being braille dots 1 and 2, it's braille dots 2 and 3. Instead of the number 1 being braille dot 1, it's braille dot 2, etc. Not exactly convenient to have to switch to the table, but once you do, you should have direct access to them.
Or at least that’s what it seemed to do, keep in mind I have very little to know formal education in rail other than non-contracted Arabic braille, which is what we get taught in school. I had to teach myself all of the rest including grade to English braille, etc.

Unless I am mistaken, I do not believe that is available on iOS.

We used to call that Nemeth code when I learned braille, and it covered everything up to calculus stuff like integrals etc, from what I remember. I mean like several large volumes of just math symbols.
Most of the small amount of it that I knew is gone from my brain, but there is a two-cell symbol to open and close the Nemeth mode.
I don't know the new braille, and I don't know the BSI stuff for the iPhone, so I have no idea if any of that still exists.

You're doing better than I, as I only ever learned the original Grades 1 & 2 of Braille, pre-UEB, that is. Still, BSI on iOS/iPadOS can do contractions and such, at least.
That makes for a nice segway for me to reitterate what I mentioned above; in that typing numbers with BSI takes an additional step.
So just be prepared for that. O:)