My concern with LinkedIn is my day job. My company is pretty active there. I have a side project SaaS that might do well with some LinkedIn marketing, but how is my boss (or more worryingly, my boss's boss's boss's boss) going to feel seeing posts from me about some side hustle? Even if the publish time is in the evenings or weekends. The alternative is the company profile route, which is fine, but basically negates my existing platform there (such as it is).
It might be beneficial for you to officially disclose your side hustle to your employer. This way you can draw the line where your intellectual property and time separate from the company’s property and time.
Consider this. You don’t post on LI. Your side hustle ends up being a success. You leave your company. There is a non-zero chance that your company will notice then, and possibly have suspicions about whose time you used to develop this successful venture.
It’s one of those situations where it is a bit uncomfortable to disclose now, but could avert a much more uncomfortable situation in the future.
My boss knows about it, we've talked about it and I've shown him interesting technical things here or there. Like I said my concern isn't so much him but higher up.
You're right that it might be a good idea to get some sort of paper trail in place. I'm already very good about what machine I do work on, what time commits happen in git (always outside of business hours, no giant commits at 5:01 PM, etc.), that sort of thing.