I don't know New Jersey law, but based on your first sentence in your post it sounds like you need to send the check by certified mail, not Zelle. Any advice you get here you should take with a grain of salt if you already looked up the law yourself. I'm sure some people have used Zelle and it is not an issue. But if the former tenant ever wants to cause an issue you will have been in the wrong. One way this can happen is if there were two tenants on the lease - the check should be made out to both of them. If you use Zelle that is probably only going to one of them, so the other person could claim they never received the deposit.
I don't know New Jersey law, but based on your first sentence in your post it sounds like you need to send the check by certified mail, not Zelle. Any advice you get here you should take with a grain of salt if you already looked up the law yourself. I'm sure some people have used Zelle and it is not an issue. But if the former tenant ever wants to cause an issue you will have been in the wrong. One way this can happen is if there were two tenants on the lease - the check should be made out to both of them. If you use Zelle that is probably only going to one of them, so the other person could claim they never received the deposit.
You answered your question before you asked it. Due to the income earned in the landlord account that goes to the benefit of the client, you should cut the check directly from the account through the bank and do as the law says, hand-deliver or certified mail. Zelle is too easy to dispute and not secure enough.
We always send deposit refunds by check to a forwarding address provided by the tenants. You always want some proof of who cashed the check and where it went.
Another reason to send it by mail is in many cases you will be deducting damages from deposit. By sending a check, in many states, if the tenant cashes the check they are de facto agreeing with your damage assessment and will have little ground to dispute your accounting. However, if you send by Zelle they had no discretion or opportunity to object and they would retain the opportunity to contest the payment. Most tenants, when receiving a check in the mail are going to cash it.