You have done all that you can really. It is up to the firm whether to decide on this for you or not unfortunately. They can say no, but typically do weigh things up on a case by case basis.
The issue is the LPC won’t necessarily give you the right to stay in the U.K as it is not a masters level course (unless you are doing a top up course) and therefore you are likely to return home between 2021-22, and therefore the work permit process for your TC becomes a little more complicated and expensive.
If the firm is not offering you a masters level LPC, one option would be to top it up to a masters level course of your own accord. You would then have the ability to apply for a post-graduate visa which would last up to two years, which would at least cover you for 2021-22 when your TC starts.
I would be completing the LPC LLM programme as the firm also thinks it's best for me to do this. Regarding the post-graduate visa which lasts up to 2 years, am I allowed to state that I would only need it for up to a year? I was under the impression that it can only be granted for 2 years, in which case I'm not sure what the process would be to switch the visa to Tier 2 in September 2022 when my TC starts, as the visa would be from 2021-2023? A little confused about this.
Also in terms of doing the LPC in 2021 - isn't the SQE going to replace the LPC then? Or would I still have the option to complete the LPC since the firm would be funding it?