Well, the tactic of simply popping in for a coffee on a regular basis to say hi does have a successful* precedent...
I don't think that the precedent of Angus and Faye is applicable. The 'stalk at place of work' part worked out for Angus. It was 'turn fascination and mutual amusement into a firm relationship' that made the whole thing collapse.
FWIW, I'm expecting Clinton to get sulky and Claire to be upset. There will probably be at least one strip this week of Claire going to Marten for comfort. Meanwhile, I guarantee that Emily will not understand why Claire is a bit cold around her for the next few days.
[Edit]
Just to expand on my sort-of-prediction for the future of this arc:
Clinton is going to be upset; he
does like Emily and this sort-of-rejection will hurt. Being human, he's going to lash out at the closest person to hand, who is Claire. He's going to accuse her of setting him up to fail as some sort of elaborate practical joke and storm off. That will upset Claire who, in her idiosyncratic way, was trying to
help him be as happy as she is with Marten. Because of this, I think she'll blame herself but, more significantly the only obvious 'malefactor', namely Emily.
Blaming Emily for anything in this would be self-defeating. I'm not sure she'd even necessarily
understand Claire if accused of doing something wrong. However, I can see Claire being a bit cold and formal with her for a while, if only because she loves her brother really and it upset her to see him hurt and for him to blame her for that. In any case, not interacting with Emily probably won't improve Claire's mood; being upset with your best friend never does. It's at that point that someone, probably Marten, will step in, mostly because he can't reconcile Claire's complaints with the Emily he knows and he doesn't want Claire to stay miserable.
Of course, I could have totally misread Emily and, with the same attitude as she manifested in strip 2999, she might instead tell off Claire for accusing her of something that she
actually hasn't done. I'm not sure if Claire would lose her temper if told she is being irrational or if the whole thing will end with a tearful reconciliation, likely at the counter in Coffee of Doom, which may lead to Dora asking them to leave such things until Emily is off-shift.