In NYC the market is used to fees for everything. As it sounds like you manage the trainers who do the trainings, your fee would be for your services as the processing agent. You provide the service of getting the client into your system, taking their information, connecting them with the trainer, etc.
I don’t find it unreasonable for you to charge a registration fee. Most trainers don’t charge fees, but they don’t usually manage much more than their own schedule.
No, I don’t charge a registration fee.
That said, sometimes there’s a start-up cost, depending on the client’s interest in doing homework, what they already have in their home, and their interests. I bring everything that we need so a client would have zero equipment cost, but some of them want homework between sessions and I no longer loan out my equipment. In initial discussions with in-home clients, we talk about what they’re willing to do during their “off” days and sometimes that requires the purchase of equipment. That can be an up-front cost of $25 for a couple of exercise bands up to $300 for a CrossCore and the mounting hardware to put it up in their garage.