Get a new bank. If the bank is a part of a significant problem for your life you should just swap it out for a bank that will let you access your accounts through a browser as needed. Toss your smart phone and only use the dumb phone.
I don't know about Ireland, but in Portugal most banks have a Web app and not just a mobile app.
You probably can't use a startup bank, as the app is one way they cut costs. Use a boring bank.
Personally I really hate that they want to kill the card, and go to an app model (with the option of a key fob). It's made even worse by the police (and others) pushing the idea that the app is safer. How is it safer to have your two-factor authentication app on the same devices as the apps that need the second factor? Those things should be separate.
I was confused by the recent change where online purchases require an SMS or the app, and no longer work with the code card.
I think the app is probably encouraged since it protects against a phishing site like this: hxxp://tilbagebetaling1060dkk.net/1060DKK/ -- but I'm not quite sure how.
Barclays don't require the app. You can still use the website or even telephone banking with a real agent. They do give you the chip and pin reader, the same as the one used in branches which works well.
Santander do seem to use a mobile for 2FA but only to send a text message and you can still use the website.
I rarely have to do more than 1 banking transaction / week so I just use the website, from home. Not too bothered about having another app, especially with access to my bank account.
TOTP isn’t exclusive to mobile apps. You can easily use the authenticator from a desktop. Some password managers (eg 1Password) have this built in as well.
My parents wouldn't be caught dead with a smartphone, but they still need to do banking.
* To clarify this, I don't mean banks that require a mobile app for TOTP or some sort of security, I mean banks that only let you operate through their mobile app. Their website is usually just a link to the app stores. and some FAQ, they don't have a physical location you can go.