Leaking fixed window help please

[EmptyNesters]

I'm cross posting from the window forum in hopes of getting some help from someone who has done this before.
Please let me know if it is improper forum etiquette to cross post.

I'm going to try and tackle the leaky fixed window that is to the right of the door as we've got a good amount of rain expected in the coming week.

I have what I believe are the needed supplies but not the know-how details. We've had our AS for about a month now so I'm still fairly green.

I could use some help please.

The window has a bead of clear silicone caulk on the outside and the gasket inside is shrunken at the mitered corners.

The process that I think I'm supposed to do is; I remove the old gasket from the inside, use a utility knife between the glass and the old butyl? tape, remove window and clean old tape and silicone.
Replace with new butyl tape, insert window, apply new gasket and then caulk the outside with what? I have TremPro 635 Polyurethane caulk and TremPro 644 RTV Silicone on hand - just not sure if either is appropriate.

How do you get the gasket in place? Just push it into the space? My fear is that I'll easily get the window out and not be able to get it back in sufficient to not leak again.

Thanks in advance for any help!!
Laurie

Photos show old gasket inside and the bead of silicone outside.