What relationship have you used to link Table A to Table B?
In that relationship, double click the relationship line linking them and enable "allow creation of records via this relationship" for Table B.
Note this works for one to one relationships. If you will have multiple related records in Table B, this layout design will only allow access to the first such related record from Table B. In such cases, a portal is usually used to enable access to multiple related records from Table B.
Ahhh... that did it and in fact, while Table A is related to Table B already via a third non-editable field, allowing the creation of items in Table B DOES allow me to change the correct related fields.
Table B has only one relation to Table A (who has many to other Tables) so no portal was necessary - Thanks for sorting that out Phil! Ironically, I had looked everywhere but there... sigh...