Looking for advice on the fastest US work visa route

I’m a UK-based graduate engineer and have been working for a UK/US tech startup as a contractor on temporary contracts since February. The company's main site is in the US and would like me to spend significant time working there. They’re currently considering having me travel between both countries while we work out a longer-term immigration solution, but there isn’t currently a clear plan for me.

I’m trying to find the fastest legitimate route to eventually work in the US, while ideally staying with the same company. I’ve looked into J-1, H-1B and L-1, but I’m unsure which options are realistically available given my relatively short employment history and current UK employment arrangement.

I’d really appreciate hearing from anyone who has been in a similar situation? I’m planning to speak with a US immigration attorney, but would appreciate any experiences or advice on what routes I should investigate first.