Unfortunately, Access does not allow extensions other than DBF for linked dBase tables. (In fact it is not Access but ODBC driver.)
Does any possibility to change the default FMM back to DBF exist in your dBase app?
BTW, why you don't use dBase to access these files? Visual FoxPro should also work but it is trying to rebuild indexes and memo files sometimes.
And the last question: Are you sure you are updating all indexes when making changes in FMM/DBF data by Access or Excel?