HF Internship while working FT?

I've had a passion for trading/investing since HS, traded just about everything that is available to a retail guy. Strategy now is just vanilla l/s equity, but been working well for past few years. Participated in student investment fund for 4 years in school and eventually ran the thing. Struck out getting a AM/HF/ER job out of school b/c of eng background, lack of fin recruiters visiting the school (well known eng school), unfamiliar with recruiting timelines, etc.

Been in a role now (

I used to make up detailed pitches in college and mail them out to HF's to show what I could produce, start up a convo, etc. Hooked a few guys, but still had trouble gaining traction. That's why I've been considering reaching out to local small HF's (decent sized list) to gauge their interest in bringing on an unpaid part-time intern.

Things I'm still running through my head and would enjoy to hear people's opinion on.
-what would other HF's think seeing on my resume that I interned at HF while still working FT? would they question my commitments or what I actually want to do with my career?
-would I need to worry about a conflict of interest with work? I'm not licensed at this job (no need of disclosing trades) and have no interaction with private side (IB, s&t)
-is this an actual productive use of time or is there something else I can do to help with building up relevant exp?

Background: Undergrad eng degree, studying for CFA level 3, outside of nyc, work at a BB in a specialized comm banking group, internships at prop trading firm and research-heavy wm shops