Clients should always have their own accounts! If a features that a client wants or needs that need a 3rd party service, I present the information to the client, explain the pricing, and let them know that the account will be a service that they subscribe to. I like to emphasize that this means that they have ownership of the service and don't have someone in the middle (like me) charging a premium as a reseller.
An example besides DeepL is a service called Smarty, which is an API to work with addresses. I am doing work with a client's CRM and explained that their address data isn't high quality enough to complete the work they need. I told them there's a free 30 day trial, and after that it's $54/mo. It sounds steep, but the client understood that it was an important necessity. It's a cost of doing business for them, just like DeepL may be.
There are two sets of pricing for DeepL, the price to use their application, and the price to use their API. There are 2 separate pages for subscription plans, and on the "Plans and pricing" page it has both subscription types. It can be confusing due to how they have it presented. Here's where they have that hidden:
I looked at the pricing plans and I don't see anything that would prevent a client from using the Free plan, unless 500,000 characters isn't enough. Then it would be €5.49 to sign up for the Pro plan, and then €20.00 for an additional 1,000,000 characters. So 1.5 million characters for €25.49. If it really came to that I would think that in most cases it would be possible to sign up for one month of Pro to translate the initial large amount of content, and then downgrade to the Free plan with 500,000 characters- unless they're publishing a ton of content per month!
In Fluency you can switch between Free and Pro accounts without any issues or additional configurations needing to be made. Hope that helps, but if not let me know.