How do Dimension Door and Find Steed / Find Greater Steed interact?

Find Steed has this sentence (and Find Greater Steed has an equivalent one):

While mounted on your steed, you can make any spell you cast that targets only you also target your steed.

Dimension Door has this restriction with regard to taking someone else with you, where the restriction is size is the important bit:

You can also bring one willing creature of your size or smaller who is carrying gear up to its carrying capacity.

Take a medium-sized character (such as a Bard with Magical Secrets) mounted on a summoned a large mount (such as a warhorse), casting Dimension Door:

Is it possible to make Dimension Door affect this mount?