Yeah I just realized this! But they only use merchant specialists. Is there a setting I can change?
Also 300 bpt? How?
I didn't do anything to the puppets. Just let them roll along (with tradepost spam everywhere). Keep in mind that I got the full Order SP track by bee-lining (not directly, but mostly beeline) to Industrial, so I had +5 hammers per city at a nice early time. This let the puppets go hog-wild building buildings. First they built the gold and happiness buildings. When that was done, they had no problem going on to the science buildings.
With only 2 "real" cities, I was able to pick up a ton of SP's. Rationalism really boosted science. I picked up later on the SP that lowers specialist unhappiness and the one that lowers specialist food consumption. Maybe those encouraged more specialists. Also, with all the economy I got going (and only 2 cities to spend it in), I had plenty of gold to spend bribing all the maritime CS (and all the rest too, if you really must know!).
I think with only a few settled cities, you can really juice your puppet economy via SP's. And since captured cities tend to be spread far apart, there's plenty of room to grow population. Add in maritime CS food and you get some good populations to build research on.
I just went back and checked my facts...I must have misremembered the 300 BPT. My capital did have 378 BPT and my other settled city about 300. My highest puppet was 239 BPT. Here's how it got there:
5 Terrain (tradeposts)
35 specialists (33 buildings and 2 from an unemployed citizen)
19 population
9.5 extra population (library)
---------
68.5 base
+50% public schools
+50% wat (siam university UB)
+100% research lab
+50% national college (counts in all cities)
--------
+250%
Total BPT = 68.5 * 250% = 239 BPT
Of my 25 puppets, 19 had 100 BPT or more, and the lowest puppet was rocking 50 BPT (my most recent additions didn't have as many buildings, but were quickly adding on). My puppets averaged above 150 BPT each
Also, note that on this particular turn I was at -1 unhappy. A few turns early or later and the research would be boosted 15% due to that SP (forget which one). From what I can tell, that 15% is on the base BPT, so another 10 BPT to the puppet described above.
So, puppets do excel at science once everything gets cooking.
In summary, to get a puppet research going, you may want to:
1 - build only a small number of your own cities - 2 is best if you can pull it off without dying early game.
2 - Beeline to industrial to get Order/Communism to get the +5 hammers per city, so that puppets can build all the research buildings as quickly as possible (and since they aren't prioritized, this is key to churning through the list of other buildings quickly).
3 - tradepost spam...

Although a decent number of farms mid-to-late game won't hurt either. Once the Order economy kicked in, my GPT went through the roof and a few less TP wouldn't have hurt me a bit. I could have rush built 15 GRDs at the end of game if I wanted (although only 2 per turn!). Ran out of things to do with my money.
4 - Horse rush to get a viable military that will capture early game puppets and have enough staying power to keep you alive mid-game. I think I diverted from a straight beeline, to get Chivalry. Alternatively, you could bypass these 4 or 5 techs and get to renaissance sooner, at more risk. I like the route I took because it gave me catapults and markets and eventually elephants (knights) and my economy needed some juice and I needed some more firepower at the time. But it wasn't optimal for the fastest slingshot.
Without Order/Communism, your puppets won't get around to the science building soon enough and your research will fall behind. You can expect with this strategy to fall behind in science mid-game until you can pop to the industrial era. The next turn, you can expect to have a 2-1 or 3-1 lead on your best rival in manufacturing and you'll catch up in science in a very very short time. After that, you've gone viral and nothing can stop you. You just need to survive the midgame science deficit.
One last note. With only 2 cities, I had a very low SP cost. By the time you bulb your GS and take the last rationalism policy for your last 4 techs to industrial, you should have enough culture left to go right to communism. If you go 3 or 4 cities, it might take you a bit longer to get there.
I got to industrial around turn 180 and I'm far from the best player. I would guess a good player could easily shave some time off this by taking a more direct tech route to acoustics.