Your call. As you’ve seen, false positives are possible with the IgG test which is why is not recommended routinely. With a single unprotected encounter, transmission is statistically unlikely.
If you really want to know and don’t want to risk a false positive, you could start with the blot, yes. Or you could start with the IgG and if you get a low positive, less than 5, you could do the blot at that time.
Can you live with not doing anything here?