We are thinking of marrying overseas then applying for a UK residence permit as the spouse of an EU national - is that faster than giving notice to marry in the UK?
It depends on the country as to what their requirements for marrying are.
If you'd had time, you could have married in the UK before your student visa expired (you would have just needed to give 15 days notice before being able to marry) and then switched to the residence permit without leaving the UK.
This would have extended your Tier 4 status so you could have continued to work and study in the UK while waiting for the residence card, and you could have finished your work while waiting for the card.
However, if your visa is expiring next week you aren't going to be able to do this.
Instead you'll have to leave the UK, come back on the student visitor visa, marry in another country, return to the US, apply for an EEA family permit, fly back to the UK and then apply for the EEA residence card!
Or, I suppose technically you could leave the UK, come back on the student visitor visa, marry in the UK and then apply for the residence card without leaving the UK (technically you're not supposed to marry in the UK on a visitor visa, but it can be possible and you are able to switch from a visitor visa to an EEA residence card from inside the UK).
Or another alternatively is return to the US, marry in the US (usually only a waiting period of a day or two), apply for the EEA family permit, move to the UK and then apply for the residence card.