Are there any fascist parties who were in the minority and did not seek to weaken the institutions of the state?
Yes. All of them. Not one of the successful fascist movements created the conditions they used to take power -the weakness in the German system had nothing to do with Hitler, the Spanish weakness wasn't caused by Sanjurjo, Mussolini didn't cripple Italy. None of the failed fascist movements in Britain, America, and France made any real effort to cripple those governments - they all sought to take over the thing intact.
The weakening of the institutions of the state is necessary for fascists with minority political power to rise to majority political power.
You can't sell a "I'm the one who can fix it" message unless the thing needs to be fixed.If the two routes to power, political or military, the Germans took the political and weakened the Reichstag though obstructive voting. With the Reichstag failing to pass resolutions and the street-level tensions constantly escalated beyond the post-WWI problems by NSDAP thugs, the NSDAP profited.
Mussolini didn't need to weaken the state institutions to gain majority political power. The threat of a peasant uprising had the rich people worried about losing their control over the government. Mussolini offered to be their strongman, taking the job of prime minister in exchange for controlling and suppressing the communist peasants.
He brought 25k Blackshirts as a dowry to the king and proceeded to use them to commit election suppression.Maybe this goes back to what you said earlier, that the fascist rise happened slower in Germany and usually doesn't take that long, but no military, no fast takeover.
You can see this in Trump's attempts to subvert the government institutions many times to lock in political majority and only then attempting the military coup. By then, his political majority was wobbling.