What was the explanation given for that? I would not even consider it unless you can sign the FT contract right now, and there is a very good explanation for why things need to be done that way. It seems to me that either they're waiting for October count (IDK if it's "October count" everywhere, but whenever they officially count students for funding) to justify the position, which means you run the risk of them not having enough students/the job not really becoming FT, or they're just manipulating you so that they can pay you a much lower salary for the first 3 months. Would you still have benefits? I don't think our long term subs get benefits.
Also, in my district anyone that isn't hired before the start of the year is automatically non-renewed at the end of the year. If they like you, they hire you back, but it's still a little stressful as they still post the position for 2 weeks. Also, since it wasn't a full year, it doesn't count for an experience increment, meaning you'd lose out on a salary step, and take one year longer to earn non-probationary status. Here, even though you were there in the same position, the long term thing wouldn't count. I would make sure you know all of the rules the district has surrounding this.
I once interviewed for a job where they told me a very similar scenario. The job would start as a long term sub but was "pretty much guaranteed" to turn into full time, and they even said I would get back pay when that happened. I still didn't even consider it- IMO there are way too many risks with something going wrong, and I needed the FT salary right away, not as back pay later.
To me this is a huge red flag. If you are just dying to get into this particular school I could see maybe justifying IF they let you sign the permanent contract right away and you ascertain there are no loopholes. Otherwise, I'd run from this position. It's not like you're unemployed and desperate to break into the field. From what you've posted, your current job honestly doesn't sound that bad and I would bet you'll discover that the grass isn't always greener.