I wanna play a Vishkanya Deadly Courtesan but my group is telling me that if I choose that archetype I can't use my racial trait because my character might be OP.
Archetypes list what abilities are gained and what abilities are lost. No racial traits is listed as being lost in the archetype’s rules, so none are lost. This is the most basic functionality of any racial archetype, so it's abundantly clear that the designers fully intended a Deadly Courtesan to use all the normal Vishkanya traits.
Perhaps an argument could be made that they were careless, but in this case, there's just not all that impressive a synergy going on between racial traits and archetype abilities; there's no ability that suddenly becomes enormously more powerful. Just straightforward, moderately powerful or even underpowered options that fit the basic concept of the race well.
I suggest you use very plain, unadorned, non-technical language to explain this to your group — but without talking down to them. If they are this skittish about overpoweredness, approaching them as though you have made an elaborate analysis of possible options is likely to be counterproductive, branding you with the dreaded label of "munchkin", "powergamer", or "optimizer" (which are often seen as the same thing by this kind of mindset). You may need to consider a more deliberate project of making sure your ideas of how the game should be played align with theirs, too.
Rules as written you would be able to use your racial archetype and your racial traits. However, if your game master says that it is possible it will unbalance the game then you must listen to him. I would suggest trying to convince him that the Vishkanya Deadly Courtesan is not OP as TuggyNE suggested, but it is possible that that particular combo will break something important to his campaign. IMO there shouldn't be anything game breaking about a Vishkanya Deadly Courtesan, but I am not your GM.