An interesting question.. No matter what, the receiver will have some sort of contact info for you. It's possible that this could be limited to the email address that you use for your Apple ID. I have not tried this, so I'm just taking a guess..
I presume the person you want to send the money to also has an Apple device that can receive Apple Cash?
You and this other person should also be using iMessage
Settings > Messages and the switch for iMessage should be on.
My thought is that you could go to Settings > Messages > Send & Receive and set the section 'Start New Conversations From' checkmark to be from your email address rather than your phone number.
Then start a new message thread with this person (I'm assuming you don't have an existing message thread if you're worried about revealing your number). I don't think it matters if you send to their Apple ID email address or their phone number. Confirm that they respond and then use the Apple Pay icon to send money
Send and receive money with Apple Pay - Apple Support
then once you're done, change that setting for 'start new conversations from' back to whatever it was, likely the phone number
Alternatively, instead of going thru all that.. you could just tell this person to initiate a message to your Apple ID email address.. once you get that message thread, use the Apple Pay icon.
Still going to want to make sure iMessage is enabled.. and also in Settings > Messages > Send & Receive, under the section 'You Can Receive iMessages to and Reply From', all the entries here should be checked.. I think they are by default. For sure make sure the email address that you give this person is checked.