Use another Google account in Chrome only
Open Chrome, tap on menu (top right corner), open settings.
On the first line, you should see your main account or google account 1 according to you..Tap on that.

You should see all the google accounts logged in to your android  device.
Below them, you will find sync option. Tap on that.

It will open a new window with title Sync. Find the syncing to option, tap on it. Now select which account the data should be synced to.

This will force chrome to use Account2 or whichever chosen account, keeping main account to the OS unchanged 

Done!!