Asking to be considered for a down leveled role. How would this look?

I'm a recent grad (EE + business minor, 3.5 GPA) with 16 months of tech work experience as a co-op student. (8 in software project management between 2 companies, 4 in business analysis for a tech startup, and 4 managing the implementation of a manufacturing hardware/software system for a F50). Also, lots of extracurricular involvement. I'm trying to get my foot in the door for PM roles as I feel it's a great fit for my profile. Locally in my Canadian city is a relatively small (~500 employees) but old (20 yo) company that's currently recruiting 2 senior PMs. I'm stoked on the possibility of a role with this company in the listed area as I hit the mark on every criteria (including fluency in a desired foreign language), however as it's a senior PM role they want 3+ years of relevant industry experience which obviously I don't have. I've found the PM manager on LinkedIn, and I'm considering sending him a quick note (100 words) introducing myself, expressing my interest, and most importantly asking whether I could be considered for a (non-senior) PM role. (With my resume attached, for context). Could this be a useful strategy, or would I be doing more harm than good with this question by coming across as pushy or unprofessional? This role has been re-posted on LinkedIn three times over the past week and each time has only received 3 - 10 applications through LinkedIn. There is some competition, but not at the level of FAANG or Tier 1 silicon valley companies.