[Google Pixel 4a 5G Guides, News, & Discussion] najabi:
I had that problem as well and found a best solution: manually download and install the same 64-bit build version of Chrome, Trichrome Library and Webview from apkmirror.com, and upgrade manually. Bromite is ok, but I prefer stable Google Chrome from Google Play to enjoy Google Sync and all goodies. I'll explain:

Starting point: I have a Xiaomi Mi 8, which has 6 GB RAM and was always frustrated that I can't install 64-bit Chrome and Webview from Play Store, because by Google's decision only 8GB up can. This is a bit ridiculous for me, because I'm losing the best performance of the main app that I use in my phone, the web browser. I don't use many apps at once and 64-bit Chrome only takes some hundred megabytes of ram more than 32-bit. And sure Chrome 64-bit is faster than 32-bit, so after lots of studing and trying I found a solution. I don't like that Google forces me to have 8Gb Ram to run 64-bit Chrome! I manage my own phone, apps and RAM. ;-)

Example for latest version at the time of writing:

I start by searching for 'Chrome Fast' in apkmirror.com. Then it links in the download page to required Trichrome Library. And finally I search for 'Webview same_trichrome_version number' and once in its download page I check it links to the same previous trichrome Library, because they interdepend. So I download in this order: Chrome Fast, Trichrome Library and Webview. Then I install in this order:

Trichrome Library 102.0.5005.99 (arm64-v8a + arm-v7a) (500509934) Notice the same build number. Mixing build numbers won't work. 34 ending builds always contain 64-bit libraries. I have examined them to make sure.

Google Chrome: Fast & Secure 102.0.5005.99 (arm64-v8a + arm-v7a) (Android 10+) (500509934 Build, always last '34' ending) (apkbundle). You need an app to install apkbundles, like APKMirror Installer. I use APP Manager (root). You can add your language if it is avaiable. Normally it installs in english.

Android System WebView 102.0.5005.99 (arm64-v8a + arm-v7a) (Android 10+) (500509934) Notice the same build number (apkbundle)

Install the three packages in order: (Trichrome, Chrome and Webview). Google Chrome 64-bit will only start when the same Trichrome Library version is installed.

Webview is optional, but is highly recommended since it is always loaded in memory and used by any application that displays a web page (i.e: Outlook). Webview also depend on the same Trichrome Library version, so you'll want all three apps in synced versions so Android is optimized for 64-bit rendering and using shared processes and memory. If you don't install Webview 64-bit you will be wasting memory for 32-bit webview processes and 64-bit process for Chrome 64-bit/Trichrome Library.

If you install all 3 components, you are actually replacing 32-bit for 64-bit rendering for the whole system, taking only a few hundred megabytes more of RAM (out of 6GB in my phone).

You'll need to install manually all three apps to upgrade, I stringly advise you to untick Chrome auto update from Google Play so it doesn't get replaced by 32-bit version again.

Important: Also, from time to time, clean your system of older Trichrome Library installations. You can't uninstall them! I haven't been able to uninstall Trichrome Library with 'adb uninstall' or any package command/app. Only solution is using a root File Browser, like 'Material Files' and manually delete folders of older versions. Each version has two places:

/data/app/com.google.android.trichromelibrary_version_number****** (~124Mb)

/data/data/com.google.android.trichromelibrary_version_number***** (a few kb)

Enjoy Chrome 64-bit! Check chrome://version page.