Often "coup d'état" is shortened to just "coup". It is not common to see coup d'état in it's full form unless you are discussing a coup d'état itself.
Golpista is a good word. It is difficult to find a direct and meaningful translation for this.
"participant in a coup" would be the most literal and direct translation. However, this phrase is so generic that most people do not normally say this.
Every coup d'état has an ideology or methodology behind it. Generally you pick a either the ideology or the methodology to describe these participants.
If you describe participants based on ideology you could call them "militants" or "nationalists" or "anarchists" or "communists". The coup itself would then be a "military coup", or a "nationalist coup", or "anarchist coup" or a "communist coup."
If you describe the participants by methodology you could say they were "extremists" or "terrorists" or "pacifist". The coup itself would then be an "extremist coup" or "terrorist coup" or "pacifist coup".
In Spanish there are words for all of these things already: "militantes, nacionalistas, anarquistas, comunistas, extremistas, terroristas, pacifistas"
So for working English, just pick the kind of golpista they are.