Do you think Ra's al Ghul should be considered Batman's archenemy instead of the Joker since Batman has described him as

"More dangerous than the Joker and Lex Luthor put together" to Superman in the animated series episode "The Demon Reborn"

and so dangerous "that he makes the Joker seem like a minor annoyance" to Nightwing in Detective Comics 700

No the Joker is his first villain and has had the most involvement with Batman in his series, excluding maybe Catwoman. Luthor is Superman's archene,y even though Darksied and Doomsday are more powerful.

Got ya now

Terrorism Trident

I would argue that presenting who exactly is the archenemy of Superman/Batman can be tricky, since these two superheroes represent a spectrum of virtues and social issues. While Lex Luthor often seems like Superman's archenemy, other formidable foes such as Doomsday, Brainiac, General Zod, and Nuclear Man also feel very worthy of philosophical consideration.

Likewise, Batman's formidable enemies such as Joker, Two-Face, and Penguin definitely create a headache for the Dark Knight, but Ra's al Ghul presents ideas about the connections between terrorism/fascism and criminality in ways that no other Gotham nemesis does.

In fact, in Ra's we find various orientations of terrorism decision-making (i.e., revolutionary mysticism) that are rather well-echoed in various revolution themed avatars in comic book literature such as Dark Phoenix (Marvel Comics), a transformed telekinetic woman who poses a serious gender challenge for the X-Men and their universe, and Cobra Commander (G.I. Joe - Hasbro), a masked leader of a lethal terrorist organization.

If we can link Ra's to characters such as Dark Phoenix and Cobra Commander, then we can argue that Batman stories about pedestrian angst about urban governance unrest would feel undeveloped if we under-value the terrorism-calligraphy of the relentlessly scheming Ra's al Ghul.

In fact, isn't the governance authority of Ra's al Ghul the rationale for linking him to the mortality-defying Lazarus Pit?

Lazarus Pit

Joker = The Archenemy. Did things to Batman Ra's never would do unless necessary. Ra's is lawful, Joker is chaotic. Like Lex Luthor, he is the one who gets under the hero's skin the most.

Ra's = Grander scope/scale villain. Bigger bad like Darkseid and Brainiac.