In Portuguese, I never found it difficult - mostly because, as the OP suggests, there is no material difference. If you want to talk about political propaganda, either say "propaganda" and let the other person deduct from context that you mean political propaganda, or explicitly say "political propaganda" (propaganda política).
In some ways, it might even be better as it will require you to actually characterise whether you mean political in the party-electoral sense, or in the ideological sense, etc.
The first time I heard the word "propaganda" in the English language, I assumed it was a less used synonymous for "advertisement". Despite having lived in an English-speaking country for over a decade, I still see them both as one and the same.
I sometimes feel like the separation is mostly used as a means to purport corporate and commercial advertising as legitimate, good and desirable (or at least acceptable) whilst keeping the idea of political and ideological advertisement as evil.
Both are bad. Both are means to manipulate an individual's opinion in favour of the advertiser. Commercially it is so I feel compelled to trade a portion of my life and health (in the form of money that I earned through work) to them in for a good or a service that I may otherwise not have thought worth the exchange.
Politically it's the same, only this time instead of my money they want my vote or my support for a certain policy that might even be against my personal or collective interests.