I like the concept of the story as a whole, and placing it after the episode "Been there, done that" is perfect. There's only one small thing: You never really specified which version of the Doctor was present. Since I am familiar with every Doctor from Patrick Troughton through to Peter Capaldi, it would be nice to definitively know which version of the Doctor our hero's are encountering. Both Tom Baker and Sylvester McCoy utilized a red scarf, (and I assumed, because of the umbrella, you were referring to the Sylvester McCoy era Doctor), however the dialogue seemed more appropriate to the Tom Baker, David Tennant, or even Matt Smith portrayals, which kind of took away from the read. When using a character as complex, and with as long a run as The Doctor, it can be challenging to nail down an individual performance, believe me, I know. Aside from that one little issue, I really like this idea. You should expand this into a full blown story.

This could ACTUALLY HAVE WORKED,with EITHER series.

But then,there would have been copyright issues,contractual snags ,and all that.

Nevertheless, it's a great story.

First off, I'm very happy you chose a Doctor. Second, well written, can't spot any errors and it's amusing. I can't offer constructive criticism because I don't see anything wrong with it.

P.S. Groundhog Day FTW.