Hi ZiksOmega and welcome to our forums!
Joomla 4.0.5 is not PHP 8.1 compatible. 4.0.6 neither, but fixes way more issues. My suggestion is to upgrade your Joomla to 4.0.6 first.
And I recommend to change hoster now if they only provide you prematurely the latest PHP without choice for the stable maintained version(s), as you will have same issue again when PHP 8.2, 8.3, 8.4, ... comes out, and that time with a site in production. PHP 8.2 will presumably introduce more changes than 8.1. Any decent hoster will provide you with a choice of PHP versions. Even on my localhost I have all PHP versions installed (using a well-maintained PHP PPA) and can switch with a single command line for my tests.
CB 2.6.x is not PHP 8.1 compatible. Some of the PHP 8.1 compatibilities improvements require to drop PHP 5.6 compatibility to remain sane, so PHP 8.1 compatibility will be introduced in CB 2.7 which has minimum PHP 7.4 requirement.
I can't give any ETA, as tests and fixes continue (as you saw it, I'm also fixing Joomla 4.0.6 and 3.10.5 at the same time, as well as adding PHP 8.1 compatibility to CBSubs. My own guess is that it will take another 1-2 weeks. But we aim to release CB 2.7 as next CB release and will probably discuss the release process next week.