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?

I don't see how it's a good idea in my honest opinion. When you sign a contract, you have a two month trial period (at least where I'm from), and the purpose of it is exactly what you mentioned. During that time, the company can evaluate your abilities and you can evaluate if you want to stay in the company.

If you try to negotiate for a lower salary, companies will just use you and negotiating for a higher salary afterwards will be hard.

Moreover, even though some internship experience is considered as 'experience' in a CV, I usually make a distinction between someone who have, for example, worked 6 months in a company as an employee and someone who has worked for 6 months as an intern. The reason for that is that I don't expect an intern to have done the same job as a full time employee, but that's my opinion.

So in the end, if you have the choice between signing for a full time job and an internship, choose the full time job. And don't worry, if you're new in the industry, they will usually take the time to train you. Don't underestimate and under-sell yourself.

Clarification: As Chronocidal pointed out (in the comments), probation periods are not mandatory for companies but I've currently never encountered a company who didn't have a probation period for new employees. And indeed, it can be more or less than two months, depending of your contract and country