“if you take the Caspian route...”
The big problem is the Russians have all of ONE frigate in the Caspian Sea. You really can’t mount much of an amphibious operation with a single, 20-year old patrol ship.
As GeronL indicated, I should have read the page 2 of the original article. The point is moot since the article does not claim that Russia is going to invade Iran.
But with regard to the limited number of warships, they aren't very much needed there. Caspian Sea is a large, often shallow lake, and it is slowly drying up. On the other hand, other countries don't have much there either, for same reasons.
Troops could be delivered on commercial ships (which are available) and sufficiently protected with air power or smaller armed boats. Don't underestimate the bandwidth of a few barges loaded with stuff.
But, as I said, the article writer believes that Putin wants to fight Georgia. Perhaps geopolitically he might want to do that, and nobody is going to ask inconvenient questions in the aftermath of attack on Iran. Israel will be probably hit by Iranian missiles, maybe to the point that this becomes a question of survival of Israel. With the obvious response by Israel.
On the other hand, Saakashvili is not a threat, currently, to Russia or other neighboring countries. He is just an annoyance. His army took losses in the previous encounter and haven't recovered yet. Georgian opposition is striving to kick Saakashvili out. Neighbors of Georgia (all of them!) hate Georgia. He has lots of problems to take care of. Putin may want to just wait until Saakashvili falls on his own.