I have purchased the Strixhaven book, have all appropriate things checked in the home page for each character, but I still can not see Silvery Barbs for my players in the list

I also own Strixhaven and cannot chose Silvery Barbs. This appears to be a problem with the DnDBeyond programming rather than an owned resources issue.

I was able to add it to a Bard character with the Fey touched feat, but not to a Cleric character with the Fey Touched feat. So it is available to me as a bard spell, but not as a Fey Touched spell.

Quote from AmblerManBill >>

I was able to add it to a Bard character with the Fey touched feat, but not to a Cleric character with the Fey Touched feat. So it is available to me as a bard spell, but not as a Fey Touched spell.

 
I made a 2014 Cleric, levelled to 4th, chose 2014 Fey Touched and could add Silvery Barbs.

I made a 2014 Cleric, levelled to 4th, chose 2024 Fey Touched and could add Silvery Barbs.

I made a 2024 Cleric, levelled to 4th, chose 2014 Fey Touched and could add Silvery Barbs.

I made a 2024 Cleric, levelled to 4th, chose 2024 Fey Touched and could add Silvery Barbs.

So I can only surmise that

Quote from AmblerManBill >>

This appears to be a problem with the DnDBeyond programming rather than an owned resources issue.

 
Appears to be incorrect and is more likely you not selecting the correct thing. 

2014 Core Rules, Legacy and Expanded Rules all need to be toggled on in the Home tab of the character. You may also need to check entitlements are in sync under Account > Licenses.