That's the way that mobile apps are designed to work. i.e. the idea is that you can stay logged in all of the time because the device you're using (such as a mobile phone or, in your case, an iPad) has its own security built into it.
For example, if I leave my mobile phone inactive for more than a few seconds (other than when, say, I'm using it to watch a YouTube video), the screen will go blank and it can't be accessed again until it sees my ugly mug (using its facial recognition feature) or I type in my PIN. So nobody can easily access my email app if I'm not there.
You need to ensure that you've got similar fairly good security enabled on your iPad, so that others can't access it without your help.
Check that you're using decent security measures on your iPad
https://support.apple.com/en-gb/guide/ipad/ipad9ae59af9/ipados
but if you're still worried, consider using BT's two-factor authentication option as well
https://www.bt.com/help/security/two-step-authentication
(I'm totally mystified as to why deleting your BT account in your iPad's email app, and then creating it again, hasn't solved the problem with your outgoing server though. In over 15 years here, I've posted that solution several hundred times and it's never failed before!)