I think it would make sense to have Dooku still be a member of the Order during TPM.
There is also another issue you run into by trying to introduce Dooku in TPM. Sure, you could tease him, but I feel like it would be strange to show him (like appearing ominously at Qui-Gon’s funeral), but to not give the audience information as to who he is. It could work well at misleading the audience into assuming that he’s the Sith Lord, only to reveal in the second film more details of who he is and why he was there. This would especially work well if Dooku is briefly seen overlooking Qui-Gon’s funeral after Mace Windu’s line, “Then who was destroyed? The master? Or the apprentice?” Snooker made a nice test of this idea forever ago and I know others have made similar things. Plus, it would be simpler to de-age him just for one shot.
This would go against canon, but I feel like this idea would pair well with Dooku not actually being a Sith Lord, but simply a fallen Jedi. Then in AOTC we are kind of on the same page as the Jedi as far as not knowing who or what to believe. We don’t have the additional context of Palpatine’s machinations.
It would also add to why the Jedi are skeptical of Palpatine in ROTS. Anakin killed Dooku, who they thought might be the “Sith Lord”, but things still don’t feel right. The Force still feels out of balance.
But simply removing the scene of Palpatine and Dooku at the end of AOTC would help this idea even if Dooku is actually a Sith and working with Palpatine.
This would also go against canon, but I could also imagine a version of TPM where Dooku was on the Jedi Council as a voice of opposition against the majority opinion that spoke out in favor of his former apprentice. But I feel like it would be difficult to make such a scene work with the material available. No real way to make it clear who Dooku is, so in a way the mystery angle that I mentioned above might be best.