I can't pair Bluetooth keyboard to my iPad iOS 10.3.3 and it is not going to update.

Notwithstanding your Bluetooth Keyboard difficulties, if your iPad cannot be updated beyond iOS 10.3.3/4, then you likely have an iPad4. This can be confirmed by your model number (Settings > General > About).

Update support for iPad4 ended September 2017. The final iOS releases for this model were iOS 10.3.3 (WiFi Only models) and iOS 10.3.4 (WiFi & Cellular). iPad4 cannot be updated to any later major version of iOS as it lacks the minimum hardware requirements needed to run more recent versions.

Many App Developers have also dropped support for older devices and iOS versions preceding iOS11/12/13 - and have withdrawn older versions of their Apps from the Apple App Store. Withdrawal of Apps supporting older models of iPad is making their utility and usefulness significantly more limited.

If any Apps that you need to use now have higher minimum iOS-version requirements, your only option is to replace your iPad with newer model. Of current models, Which all support the most recent version of iPadOS (iPadOS 13.5.1 is current), even the entry-level iPad7 has hardware specifications and capabilities that will vastly exceed your old iPad4.

I hope this information is helpful.

What Bluetooth keyboard brand/model is this?

What happens when you try?