Which licence do I need for client work?

Buy the licence that matches the project you are building, and hand it to the client with the work. What matters is one project holding one licence, not whose card paid.

Which tier

Decide on the client's finished site, not on your invoice. You charging the client to build it does not make it Extended. If the site itself is free for the public to use, Regular is correct. If the client will charge their own users, that project needs Extended.

Transferring it

A licence can move to the client when you hand the project over. The clean way is to tell them the purchase code belongs to them now, and to keep a record of the handover with the rest of the project documentation. If they need the licence under their own account, open a ticket and we will move it.

Buying on the client's behalf

Two ways, both fine. Buy it yourself and pass it on, which is simpler while you are working. Or have the client buy it on their own account and give you the code, which is tidier if they will be maintaining the site long after you have moved on. Agencies running many projects usually find the second saves untangling later.

One project each

Five clients means five licences. You cannot buy one licence and reuse it across the client work you take on, even if you never charge for the item itself. See one licence, one project.