Hi,
If you have two Google IDs and one is your Apple ID you can set up three accounts in the Messages App (if you so wish).
The Google Apple ID one should be used in the iMessages account using the Password you use at Apple for this ID.
For this it does not need the Less Secure App setting at Google account's settings. Your Apple ID (the Google one) should be using a password you set it up with at Apple.
However you can also use this Google ID to set up a Google Account in Messages and this will need the Less Secure App setting at Google as it is actually using the Google Talk server and needs to login to Google and does so using SSL (Which Google have deemed as "less secure").
Your other Google ID can also be set up as a Google Account in Messages - it too will need the Less Secure App setting at Google.
IF the iMessages Account looks like it is logged in but is retuning "Not Delivered" for every single iMessages that is sent then it is not logged in to the iMessages severs properly (If you use the ID in iTunes and other places across Apple it is likely to still work - this is about the iMessages servers use of the Apple ID and the Mac's Login with it).
You can try going to Applications/Utilities and opening Keychain Access and finding all the IDS related items and deleting them (Have the Login item selected top left and start with Passwords lower left - you may have to also check "certificates".)
After this restart the Messages app.
The IDS items should be recreated if the app gets to login properly (if the Keychain was at fault).
if this does not solve the issue then you will need to contact Apple.
You will need to explain that you are sure you have an Apple ID issue with the iMessages Servers regarding your Mac's use of the Apple ID.
Level 1 people at Apple Support can be very Script led and you may need to ask for a Level 2 person.
They may still insist on some checks being run.
Apple ID/Account issues like this do not need Apple Care and should not Cost.
7:41 pm Thursday; April 16, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad