Transfer your domains to Google Domains. Use the free email alias feature to rewrite your custom domain users into Gmail users (or wherever else). Configure Gmail to send from your custom domain.
Can you really configure to send from the forwarded address? From the faq[0] page: "When you need to both receive and reply to emails from a custom address, and unlock other professional features, upgrade to Google Workspace."
You might want to go over some of your email/message/security headers and look for anything unexpected (e.g. actual email address, device names, locations, etc.). The web-interface won't display most of what you'll find in there, with good reason.
Finding a way to access those headers from gmail will be a challenge though. Don't think I've ever found that button in the same place twice!
I don't think that this will enable you to send emails that conform to the DMARC standard (using DKIM and SPF). Might be ok for a side project, but in my opinion not enough for a small business.
My domain is still set up with G Suite with SPF and all so Google is technically fully capable of sending emails on my domain’s behalf, but if I got the "me@example.com via example@gmail.com" text regardless.
My setup:
1. log into example@gmail.com
2. set up "Send as me@example.com" with alias
3. enter example@gmail.com as the SMTP username
Headers:
ARC-Authentication-Results: i=1; mx.google.com;
spf=pass (google.com: domain of example@gmail.com designates 209.85.220.41 as permitted sender) smtp.mailfrom=example@gmail.com
Return-Path: <example@gmail.com>
From: Me <me@example.com>
I don't know other registrars offerings, but OP was particularly concerned about using Gmail (or something like it) and having good deliverability, which makes Google Domains + Gmail the right solution in my mind.
Transfer your domains to Google Domains. Use the free email alias feature to rewrite your custom domain users into Gmail users (or wherever else). Configure Gmail to send from your custom domain.
I do this. It works spectacularly.