Just so you don't forget if it resorts to it for any reason:
- Textnow.com allows you to sign up for a phone number on your laptop using an email. It works cross platform
- Chatango is good for pmming. You can chose to delete or save messages, and all of it is private. However, strangers can message you so i'd recommend not filling out the profile.
- There's an online web browser version of discord, download free and you can log out after every use.
If I remember any more, I'll let you know.
Oh! also, there's this website I've been on since i was 9. It's a pokemon game but it allows private messages between users. It's fun, and it can be a way for you two to talk.
https://pkmneclipse.net/