IMO, you're overcomplicating things a lot. You already have a working mail server, which is not easy to setup and maintain. Adding another mail server and redistributing mail storage / routing mail between the two will drastically decrease stability and may affect deliverability of your mail as well.
Why not just add storage to the existing VPS? If your hosting provider doesn't provide such option, you can mount third-party S3-compatible storage (rclone mount works great, a few hundred Gb on Backblaze cost virtually nothing) and store your mail there.
You call it "webmail client", but if it stores email and serves it to other clients - it's already server anyway :)
And you mentioned you need to "check/send email from any device/mobile". You're probably not going to use webmail from mobile (it must be a disappointing experience)... so you need IMAP on this "client-server" too. At least you can keep SMTP part unchanged on your VPS and send via it directly... But if you have something like ActiveSync (SoGo etc.) - things get more interesting. Even without ActiveSync, CalDAV/CardDAV also can be an additional problem if you need to archive old calendars as well as mail... etc.
Considering all these nuances that may arise in course of implementing your idea, I've suggested an easier way.