Google provides Chrome only via the Play Store. There is no legal way of installing Chrome on a Non-GMS Version of TC75. You need to use the GMS Version of TC75 which comes with Chrome and Google Play.
Zebra provides a free utility called StageNow which is part of Mobility DNA suite of tools for Zebra Android devices.
With StageNow you can push and install any apk file onto a TC7x device thru Wi-Fi.
And yes you can run Chrome browser on the Enterprise OS version (without GMS) of TC7x.
if you want 'enterprizey' soluton for non-'enterprizey' browser I would recommend MDM (SOTI or similar) as you could control access to playstore (only official way to get app and updates) and also install app remotely (but I'm not sure about 'enterprizey' way of unattended remote installation of app directly from play store without setting google account on device)