As far as I'm aware, though I could be wrong (and I hope I am because then I can get a new laptop for work!) you can't claim back expenses for AE work. What you are paid is what you declare, no matter what you have spent, which is why it is a 'simplified' business model. So as far as I can see, you'd have to cover the costs yourself and pay tax on them. Not very fair, but there you go. You'd need to change to a more complicated fiscal regime if you have high overheads and want to claim parts of it back. Unfortunately, with that comes a whole host of other issues.
Builders who have expensive overheads and material costs, for example, might ask the client to provide materials themselves, as it is not tax deductible as it is in the UK. Otherwise, they will be paid and taxed at almost 20% on their gross earnings, which means they are paying tax on what they have bought.
I'd be interested to see if anyone else has a different take on it. I've read the AE stuff backwards and have been AE for a couple of years, but as far as I'm aware, that's why it's a 'régime simplifiée' and why it's so easy on the paperwork, because it's not very friendly, if you have complicated financial operations. PL are also for people whose asset is themselves and don't need as much by way of equipment, otherwise other business models would be more suitable.