Hesitant About Posting to LinkedIn About My Side Gig

I have read a lot of positive things about using LinkedIn to promote your product pre-launch.

I am building Big Picture (https://bigpicture.site/) "in public" and was considering linking to, or even just outright copying, my blog posts (https://bigpicture.site/blog) into LinkedIn as a way to get some pre-launch traffic and some people signing up to the mailing list to be notified when we launch, and to be offered an opportunity to be an early adopter for a life time discount.

However, I am EXTREMELY reluctant to post about this on LinkedIn.

I have a full-time job and do not want my current employer to be aware I am waking up at 4am every day to work on this side project. I don't want them to question me about the time I spend or the work I do. I don't want them to try and find an excuse that this project might be a conflict of interest (I don't believe it is, but who knows what their lawyers could justify.)

How do you guys deal with this?

Is it advisable to start a new, blank, LinkedIn profile without any of my existing contacts or work experience?

Or, if these things are truly concerns for me, is it best just to not try to use Linked In for this purpose right now?

Thanks!