I'm going to say no.
Firstly, on this IMDB page, some of the differences between the versions are listed. The scene you refer to is described:
In the original television version, there is more dialogue in the "no
shirt, no shoes, no dice" scene. Spicoli says "I have uno
nickel-ette...and a pick". He then makes up a story of how Mick Jagger
gave him that pick.
However, a quick pop over to Amazon shows one of the highest reviews saying the following:
I can't believe the director of this movie keeps standing for the out
and out censorship of this! It's (at least) the third release of it on
DVD, and this latest (Nov. 2004) try is still a dud. The big problem
remains the scenes which they only include in the cable TV versions of
this, but keep cutting from the DVD. Why the heck do they?
One of the missing scenes is a very cool safe sex/birth control scene
in the mall, for instance. Fast Times sticks out as one of the very
rare films that this odd censorship has been performed on. In fact,
out of thousands of releases since DVDs started in 1995, this is
almost the only one this has ever happened to!
So again we can only ask why? WHY?
Now, that review was written in 2005, so is nearly 9 years old. But a quick traipse through the comments shows that discussions about the review have been raging ever since, with the most recent actually being posted in July 2014. The comments are mostly written by fans of the movie, such as yourself, who are wanting to know why extra scenes have not made it in to the DVD release.
There's also quite a few other forums online (e.g. here and here) to name just a very few where discussions have raged over what is different in the versions and why they haven't been released.
Given all this discussion, it seems certain to me that the full version, with all scenes intact has never been released.
On a final note, it may interest you to know that there is a petition going to have the full version released - unfortunately, as the time of writing, it only has 16 signatures...