Marking cards as 'not for trade' in Pokemon TCG Online

In the Pokemon Trading Card Game Online, when creating a trade request, the "I am Giving" option will by default only give you a list of cards currently marked as "for trade". In order to view all cards that are not trade-locked, one has to modify the filters.

When browsing open trade requests from other players, if I have no filter applied (apart from 'acceptable trades only'), results will display for all the trades I could fulfil based on all the cards I own that are not trade-locked. It appears as if the for-trade property is not factored in.

There are some cards I own - such as Tapu Lele-GX and Gardevoir-GX - that I own and are not trade-locked but seem to be in very high demand. Any time I search open requests, a majority of results will involve players asking for these cards. My question is; can cards be somehow labelled as 'not for trading', or can a filter be applied to only show acceptable trades with cards marked as 'for trade'?