Well, the smallest,lightest, and LOWEST-COST telephoto under 300mm AND with a low f/number is probably going to be...drum roll....a used manual focus 135mm f/2.8 lens. If you shoot Canon EOS, then you can choose from nine different lens mounts (like M42,Nikon,Yashica/Contax,Leica R,Olympus OM,etc) and get a $10 Chinese-made lens adapter that will allow you to mount the used lens onto a new EOS digital or film camera. 135mm f/2.8 telephotos were always plentiful, and are easy to carry, easy to manually focus, and easy to SHOOT. The focal length is not too big, and will easily be overlooked by security guards, boyfriends, insane dads, and mom-zillas...
Shoot...then crop later. 135mm/2.8 used manual focus lenses can be picked up second hand (fifth hand these days is probably more-accurate!) for $10 to $145, depending on brand. "Off-brand" 135/2.8's typically sell for only $10-$30 these days. I bought one a while back at Goodwill for $19.99, with a set of three extension tubes mounted to it!
If you have, say, a "Baby Nikon", they will accept all the old, pre-AI Nikon mount lenses,unmodified, so say a used 135/2.8 Q.C. Auto-Nikkor (vintage of oh,say 1973 and earlier) is like $60 these days in great shape! Vivtar used to sell a 200mm f/3.5 manual focus lens very affordably, and there are a number of those around these days for $40 or so.