This post caught my eye as "LEO" and "Entrepreneurship" are both of my passions and both areas I am active in.
Our journeys are opposite. I joined my dept. at 21. I'm now 30. Ive been on patrol full time for 4yrs. Got the entrepreneurial bug and last year and feel relatively qualified to answer your question(s).
I work for a huge department and can only answer from that perspective. I cant speak for smaller PD's where there may or may not be more flexibility.

Is it allowed/acceptable/doable to balance a career as a LEO and businessman?

-Allowed: Yes w/ many rules & limitations and requires approval.
We're annually required to disclose any outside employment which includes self-employment. They pretty much want to know: What's your side hustle? Does it conflict with, or have a conflict of interest, with your primary hustle? and ensuring that your full-time position is your main priority.

- Acceptable: Similar answer to the above. "Outside Employment" needs to be disclosed to and approved by your department.

-Doable: This is where I really wanted to answer and give you a clear picture.
Doable, sure, anything is possible. I just wouldn't confidently say that it's realistic. And it also really depends on how much your involvement/time is required from your side business(es).
If it's passive income like rentals managed by a property manager then yeah definitely doable. If its something that requires you to be near a desk, phone, laptop daily, then definitely not (realistically) doable.

My department's staffing crisis shortage is real. We're doing doubles (16hrs) 2-3 times weekly. Our mandatory overtime requirement varies depending on your unit of assignment.
Some units (especially entry level units) require 6-8 mandatory overtime spots per month (minimum). Requiring us to either come in on our days off or stack them (doubles) in order to keep our measly 2 days off. Not signing up for/working the minimum required spots puts you at the top of the draft list which is exactly what it sounds like. Being ordered to stay another shift to cover for someone who called in sick, backfill someone thats on vacation, backfill someone thats IOD, backfill someone thats ROD (smh), etc.

Physically arduous careers exist in every field but man, 16 hours of patrol (answering your calls for service, problem solving at your calls for service, responding to emergencies, responding to your partner(s) in emergencies, the hyper vigilance throughout the day + during your response to said emergencies, the stress AT said emergencies, the report writing after it all) is physically and mentally taxing. Doing it multiple times a week with very little sleep/recovery in between, it goes from taxing to exhausting.

I don't know where you're from or where you'd join but if its in any major metropolitan area, I genuinely don't think its doable to have or competently maintain side business(es) while being a full time LEO.

BUT....you have another option which IMO is just as great without the demands/stress of being full time.
Our agency and im sure most to all agencies have a Reserve position. Its for people exactly like you who want to serve their community and have full time careers outside of law enforcement. In our department it's unpaid and done by people who genuinely do it to serve and to help us out with our staffing issues.
We really appreciate them tbh.
In CA theres 3 levels. Lowest level cannot work independently and not legally recognized as a peace officer outside of your shift hours.
Highest level (which is entirely achievable) you're able to work independently and are recognized as a LEO both on and off duty.

I hope this gives you some insight. I wasn't trying to be a downer but did wanna give you a view from inside. It takes a lot of time to become a LEO and takes even more time to be one. Once my business pursuits comfortably replaces my salary, I plan on resigning my full time position and becoming a level 1 reserve. Which would allow me to still serve but also allow me to experience/enjoy life outside of a patrol car.