Change the same work from contract to AE

Apologies if this is a complex question - I will try to explain as fully as possible!
In summary I am trying to work out if there will be problems moving from an employed status for a part-time job in the UK to doing this work on an AE basis. This fits the employment type better, but also makes taxation/SS payments simpler and more appropriate.

I have recently set up as AE in France. I am resident in France and am also employed part-time in the UK for one main job and one ‘additional’ job. I travel back to the UK regularly for my main job and this won’t change. For my ‘additional’ job in the UK I am currently contracted to 7 hours per week (paid as a monthly salary) but the workload is highly variable. I review scientific papers and some months there are lots of papers to review and some months there are very few. I can do the work when and where I want and I use my own equipment. According to UK (HMRC) rules my work pattern fits better with self-employed status. I would like to change this to AE status - partly because this is more accurate but also because I am taxed in the UK (through my salary) and also paid tax on this in France for the same work from when I moved here in August 2020 (this is complex and I have a UK and French accountant!). The work is done remotely, and the vast majorty of it is done while I’m in France.

I also do work on an AE basis for one French and one UK organisation currrently so there is no issue with only having one AE customer.

My question, then, is are there any issues with changing from an employed status to an AE status for essentially the same work. I.e. I would change from being paid through a UK salary to invoicing the organisation I am currently employed by. But the amounts would vary a lot each month based on the workload. To clarify, it’s more that the contracted/employed status doesn’t fit the workload rather than trying to make it fit into AE status to make it easier!

Apologies again for the complicated post - this service has been so helpful in setting up my work here!

Many thanks

Jonathan