"---pete---" you said
Question is, how can we make our own brine that is
similar to the leftover store bought pickle juice, to be
used just for making quick pickles to be kept in the
refrigerator for a month or so?

Try this recipie you should be pleased
Pickles Quick Dill
1cup white vinegar 1/2 cup sugar
4 cups water 1/4 cup kosher salt/sea salt
Bring to boil and taste for sweetness/saltiness and adjust to your
taste.
Let cool to room temp.
1 TBL whole mustard seed 1TBL mixed whole spices (pickling spices)
Garlic cloves and onion(your personal amount) about 8 or so grape leaves
Dill (again your personal amount)
In a gallon container pack the cukes in tightly(sliced or whole with the
dill/garlic and grape leaves.
Add the bine which has the spices in it and let cool completly on counter
(so you dont crack the container)than refrigerate.
Pickles will be ready in about 2days

This may not be EXACLTLY like store bought but "Try It You'll Like It"

Your cukes get mushy as they need alum to stay crunchy. Grape leaves have
alum in them so use grape leaves instead
any type will do .I often use the wild ones growing on the edge of our
field.
Folks who try this please let me know how it came out for you.

Jerome