Yes, so would I be ashamed, but I don't think it should reflect badly on LFC. A lot of its supporters, certainly, but not the club.
Unless it comes out defending him, of course.
It'll reflect badly on the club regardless of what they do to some extent simply by association as a Liverpool player, that's to be expected of any individual representing a corporate body (disgraced MP or executive for example, individual action always has repercussions for the organisation). Uruguay have already blundered by leaping to his defence, LFC haven't made an acknowledgement yet. That is somewhat understandable, he's away from the club and they'll need to draft a corporate line on this which will probably mean legal and PR advice as to how to address the incident.
It may be unfair on the club to but that's the rules of engagement. And look at the history here:
Suarez cheats in a World Cup, then bites a player in the Dutch league, he is respectively sent off and banned for 7 games.
Liverpool (class and dignity FC) buy him. He is found guilty of racially abusing a player, they leap to his defence with a ridiculous PR stunt, he is banned again.
He refuses to shake the offered hand of the player he abused, and is defended by his then manager for that action.
Then he bites someone in the premier league in a high profile game and is banned again.
The club stick by him and he responds by trying to force a move, is then told he should apologise by his manager or he won't be played, he doesn't but is played anyway.
He is venerated by the English press and made player of the year, then goes to a world cup and goes on an offensive against the same press after knocking England out, all the while insinuations are rife that he is leaving Liverpool.
Then he bites someone again during the biggest game on the planet at that point while still a Liverpool player.
There is no way this isn't going to be bad press for LFC.