The best way to achieve this would be to use the FM tone generator in Sound Forge. Lacking that and assuming you already have the 1KHz tone do this.
Add the Elastique FX to the track header. R Click the track header and select insert automation envelope. Select the Elastique FX and tick the pitch ratio box. Add nodes and adjust envelope as needed to get the effect reuqired. You are limited to +/- 2 octaves though. Give that it's a sine wave you could render out and repeat I suspect without anything falling apart too badly if you needed a larger pitch shift.
"Adding vibrato to other sounds creates really weird stuff"
That's the advantage of the elastique plug, best in the industry when it comes to artifacts.
I did send you an email so I could send you a project file using that FX, maybe your contact details are out of date. You can ping me if you still want it.
However on further thought maybe you don't have that plug.
I know V9 and V10 use the elastique technology for time remapping what I'm not certain of is if the actual plugin is also included. I also have SF10 and I may have 'acquired' that plugin in Vegas from SF10, don't know for sure.
If you'd like to be technically accurate, the normal IR tracking tone of the AIM-9M Sidewinder is a steady 1500 Hz, not a warbling 1000. In the newer version (AIM-9M-9) there are several different tones to indicate the Sidewinder is seeing (or thinks it is seeing) flares from the target aircraft and whether it thinks it can "hack" the IR countermeasures (flares). I forget the specifics on the different tones, but I'd suspect you're just looking for the normal lock-on tone anyway.