Yes, “variety” is singular", so “there is” is correct. But you could say “There are many different options”, because then “options” controls whether you have singular or plural is/are.
Well, in my humble opinion, in this case Fowler is wrong. (He’s almost always right, though sometimes a little oldfashioned these days, but even he can get it wrong on rare occasions),
I think it is just the “a” coming directly after “are” that seems to grate. However, I would have no qualms about saying “There are a lot of options”, or “You are a knucklehead,” so I don’t know why that should be. Anyway, A it is.
It seems to me that “a variety of” could be replaced with “seven” or “a dozen” to give a more accurate picture of the sentence. In this case, it would be
“There are (a variety of/seven/a dozen) options.”
However, I think you can be granted some leeway here, since “of options” could also be construed as a prepositional phrase modifying “a (singular) variety”.
There is a regiment (of soldiers) on my lawn?
or
There are (a regiment of) soldiers on my lawn?
Since the subject is “a variety”, I’d take the singular verb.
However, group-nouns, like number, regiment, etc. are funny. In English English, for instance, the government are… in American English, the government is.
So it seems that there is no concensus. However, I don’t think the agreement by proximity is a very good rule. Consider:
“Either John’s brothers or Sally is/are bringing the dessert.”
In my opinion, both options are awkward. Despite the length, I would prefer:
“Either John’s brothers are bringing the dessert, or Sally is bringing the dessert.”
On the thing with “variety,” the best choice, if grammar is your chief concern, is “There is a variety of options for that car.” 78% of the panelists consider that to be a good choice.
However, in casual writing, it might be better to go with whatever sounds best. Consider:
“There are only two options for the Civic, but there is/are a wide variety of options for the Accord.”
Here, using parallel sentence construction might tip the balance. Still, technically, “is” is correct.
All that said, I would prefer “many options” to “a variety of options,” just to avoid this awkward problem.