As far as I know, yes, you can pay for your Pro subscription and use it for commercial projects at the company.
Unfortunately it is hard to find a clear list of what you can or can’t do with a subscription and I am not a Docker staff member to give you an offical answer, but I quote from the FAQ:
I work at a large company (5,000 employees, $2 billion annual revenue) and have been using Docker Desktop for free. Under the new subscription terms, do I need a paid subscription to continue using Docker Desktop?
Yes. Commercial use of Docker Desktop at a company of more than 250 employees OR more than $10 million in annual revenue requires a paid subscription (Pro, Team, or Business) to use Docker Desktop. While the updated terms for Docker Desktop are effective as of August 31, 2021 there is a grace period until January 31, 2022 for those that need to transition from a free to a paid subscription to use Docker Desktop.
I highlighted that it says you can have Pro licence which is for individual developers.
Also notice the conditions:
more than 250 employees OR more than $10 million in annual revenue
So if you are working for a smaller company with less profit, you may continue to use Docker Desktop for free. But I guess the company wanted to use an alternative because otherwise they should have paid for Docker Desktop.