Short answer: no. there's no way to make page-change events trigger any other event in MIDI Designer.
HOWEVER: you could get a page-up/page-down to combine with the send-bank-info (so you would have the sequential page-up-then-send-bank-info). Basically you'd need a supercontrol button that has two subcontrol buttons, the first of which is a page-down and the second of which sends your bank info.
See: Multiple MIDI Messages in MIDI Designer.
Note: This is a (pretty kludgey) workaround, though I have added the suggestion to be able to send out midi messages in response to page selection.
Thanks for your question. Best!
Dan