Well, maybe. The rules are a bit ambiguous on this point although in your specific case of domestic LA and LP flights they appear to be on your side.
From the BA RTW rules;
<font face="Verdana, Arial, Helvetica, sans-serif" size="2">
Booking codes .......... First/Concorde Business Economy
AA AY BA CX IB QF LA EI ........A ..........D...... M
AY EI LA LP DOMESTIC ...........Y ..........Y ..... M
BA WORLD TRAVELLER PLUS ........................... T
</font>
That says (to me) that if you have a DONE4 ticket and fly LA or LP domestic then you fly Y -- even if there is a business or first cabin available.
However, also from the rules:
<font face="Verdana, Arial, Helvetica, sans-serif" size="2">
For flights where no first or business class is available or offered first class passengers will travel in business / economy class and business class passengers will travel in economy class. The fare for the highest class used applies and no refund will be given.</font>
Now that reads like you will fly "economy" and that is designated as "M" for OWE tickets in the preceding chart
Anyway, I do agree on the agent's lack of intimate knowledge of the rules and hope you won your argument.