Let's say that Jaime Reyes, Paco, Brenda and other BB supporting cast members
from Blue Beetle volume seven had lived in Gotham City and it was shown in Infinite
Crisis #3 that the Blue Beetle scarab came to earth in Gotham City's district area
dubbed "El Paso". 
 
Let's say that it was there that Jaime found it and took it to his house.

 
And let's say that things happened as before, despite the different setting.

Now, what would happen if Jaime Reyes appears miles away (and at the start of the seventh Blue
Beetle series) from Gotham with no memory and clothes one year later after Infinite Crisis?

What would happen if Jaime became a hero in Batman's home turf
and Bats finds that the kid is not only battling one of Gotham's most dangerous, but
also other villains that he'll have to battle to defend Gotham (and who are more or less connected to The Reach plot point)?

How would Robin and the rest of Bat-Family, besides you know who, react to Jaime performing his heroic duties in Gotham?

How would Harvey Dent (posing as Gotham's protector during Batman, Robin & Nightwing's long absence)

react to Jaime defending the GC?

Would the new Blue Beetle get involve in any, some or all of the Batman storyline events in
his series and/or in the Bat-titles? Would he do anything to effect those events?

And for that matter, would the seventh Blue B. series be billed as a Bat-title by everyone since
Jaime, his cast, villains and adventures are based in Gotham City or for another reason (if that'll even happen at all)?