I would suggest you write on substack/medium and post about it on LinkedIn. It will give you visibility among the VC community.
Investment thesis is just your take on what you think about the business. I don’t think your employer will think you are switching jobs, it would be more like you are analysing and posting your views.
Being anonymous won’t help. Also, VC is a job of 14-16 hours, it’s not all hanky dory, you don’t advise founders or seat on board seats. Most of the time you are just researching or sourcing and convince the partners to invest(very high scrutiny). It’s a lonely job, so be very sure to invest time.
It will almost take 6 months of hard work to get into it