Bingo, Dmills! Thanks for answering. That is exactly what I am trying to do. Disgusted by the awful pricing of "Video SFP" modules and even commercially available converters, I wanted to try just using a generic, ultracheap 10Gbit SFP than can be had for $8 and shove the 3Gbit SDI data straight into it.
So, what I did, is just plumbed SDI receiver and transmitter chips behind the SFP module, in the hope it will work as a proof of concept.

Rest of the stuff is just an MCU so I can play a bit with the control signals and make an EEPROM reader (writer

) for the SFP modules.
But I have not used any reclocker chip. Could you please elaborate more on this topic? I am no expert on hispeed digital design, just a beginner bit wrangler and would like to learn some more.
Looking in the SDI standard (SMPTE 424M), they specify maximum timing jitter of 2 UI and alignment jitter of 0.3 UI. Not sure what that means exactly (need to study up). I tried to find jitter spec of SFPs, but this is not available to most modules, especially cheap ones.
I've only found one datasheet, where a jitter of 0.07 UI peak-peak and 0.007 UI rms is specified for the transmitter, but unfortunately no spec for the receiver.
My uneducated guess is, that it may work without a reclocker just fine, but if I put the data back into a coax cable, I decrease the maximum achievable distance then and may be even out of spec of SDI? Jitter on Gb data is unfortunately not a thing individual can measure at home.

So, on which side would you use the reclocker? I think it makes sense to use it on the RX output of the SFP. Do you have any specific recommendation of a chip? I know only about some LMH0318 or what, pretty nasty expensive.
//EDIT: LMH0346 .. jeez. That pricing is mad. LMH0366 would be especially suitable, as it has a data rate auto detect and supports MADI.. but oh noo the price. Okay, let's see what Semtech has: GS3281 .. ughh, they seem the need to compete in screwing the pricing.
//EDIT2: Macom also does lot of reclockers, but datasheets under NDA? Srsly?