Is it ok to offer lower paid work as a trial period before negotiating for a full-time job?

I am currently employed as a TA (statistics related) at a university until September, but I'm looking to transition into industry.

I am looking to work in software development or data science. I already have some working experience (several internships let's say) in IT, mainly web development.

I have been doing programming for almost ten years now, but not much professionally so I wouldn't really class myself as a junior, but also definitely not as a senior yet.

So I thought it might be a good idea for me when I apply for a job to offer to do like a month or two internship, with lower pay (I wouldn't need money now, as I have paid summer off as a TA), so the company can get a real world idea of my abilities and how I'd fit in with the team before discussing full-time employment and salary.

It seems like a win win scenario, as on the one hand I'd gain some more experience and catch up with the developments in the industry (having been in academia for some time now, I am a bit rusty, but can get up to speed quickly) and on the other hand, the company is making a much lower risk hiring me after the trial period, as they know much better and there is much less uncertainty, relative to straight up giving me an offer.

What do you think about this?

Is offering low pay work for a trial period over the summer before discussing full-time employment a good idea?