Are there any linguistic languages where the goal is to be difficult to speak, write, and understand, while still being actual languages?

Tolkien is known for writing languages for his books, specifically elfish. But instead of drafting a dictionary for elfish he used his knowledge as a linguist to create rules that allowed for 'reconstruction'/generation of words.

In contrast, Orcish/Black Speech, was specifically left fragmented, using harsh unpleasant sounds, and agglutinative grammar that changes meaning with context instead of adding prefixes/suffixes like English or Elfish.

Tolkien also never wrote songs, poems, or stories in black speech. The inscription on the One Ring afaik is the most complex writing in black speech.

So in a sense, black speech, when juxtaposed with the intuitive design of elfish, kind of fits what you're asking for; Albeit it's likely not a functional language.

Klingon. Most of the emphasis is on being difficult to speak, but they also moved the word order around for maximum alien-ness.

It might be stretching the definition of "actual language" since it's fictional and there's no formal framework for it, but my first thought was the Marklar language from South Park. It's the linguistic equivalent of the Whitespace programming language: every noun is Marklar. To the South Park kids it's completely confusing, while remaining perfectly intelligible to the Marklar.

Navajo code used by the USA during WWII was famously unbreakable, if that counts

Oh that would be a very fun project! Make every sound just a different kind of vowel, skip all the consonants entirely. Make sure each word has vowels that are very similar. Every word should be very long as well.

Everything should have conjugations, and you should be able to stack them infinitely. There should be lots of prefixes and suffixes you can sprinkle liberally everywhere.

The grammar should have zero consistency. Everything is just a never ending list of exceptions. You would need to have a degree in cryotherapy to learn this language.

The classification of words and linguistic structures should be so bizarre, that existing systems wouldn’t be able to handle it. None of this fits in any box we already have. It should be so unique, that no other language would help with it.

Ithkuil wasn't made to be hard, but it was made to express "more profound levels of human cognition briefly yet overtly and clearly', and as a consequence became very hard

Khoisan languages might be in this category.

https://en.wikipedia.org/wiki/Khoisan_languages

Khoisan languages are best known for their use of click consonants as phonemes. These are typically written with characters such as ǃ and ǂ. Clicks are quite versatile as consonants, as they involve two articulations of the tongue which can operate partially independently. Consequently, the languages with the greatest numbers of consonants in the world are Khoisan. The Juǀʼhoan language has 48 click consonants among nearly as many non-click consonants, strident and pharyngealized vowels, and four tones. The ǃXóõ and ǂHõã languages are even more complex.

Probably one of the submissions to Agma Schwa's Cursed Conlang Circus.

Seraphim nearly fits the bill.