Nuclear fission products include a lot more than just Sr-90. Yes, Cs-137 is a beta emitter. But it decays to a metastable isotope of barium which gives off a characteristic 662 keV gamma, so Cs-137 should be considered both a beta and gamma emitter. Cs-137 is going to be found alongside Sr-90 in any nuclear fallout. Today, both Sr-90 and Cs-137 are in ample abundance in the Chernobyl Exclusion Zone. The Mayak accident released significant Cs-137. It is not possible that any significant products of nuclear fission would be beta-only.
I need to note that Geiger tubes are not terribly sensitive to gammas; they have a poor gamma detection efficiency. By comparison, Geiger tubes that are sensitive to betas typically have very high counting efficiency. So I can't completely rule out fission products. It is possible that the clothing is just barely hot enough with betas to be easily detected, but not hot enough for its gamma activity to be detected by Geiger counters with limited gamma efficiency. A more sensitive gamma detector, like a scintillator, would easily detect gammas that would be expected from fallout that also produces this much beta emission. So when the radiation was classified as beta-only, I am very interested in what instruments they used to rule out gamma. I would say fission products are possible if and only if the dosimetry lab had poor gamma detection instruments and the clothing fell in a rather narrow window of activity. Still, I don't see it as very likely.
You can actually purchase the same Geiger tubes used to measure the clothing. I interpret "Recalculation was performed on the B-2 installation in a lead house with a cassette counter STS-6 in the number of 4 pieces." as meaning a counter with 4 STS-6 tubes surrounded by lead to cut down on background radiation. (The English term is "lead castle.") STS-6 tubes show up on eBay. They are sensitive to gamma and hard beta. Here is a datasheet (in Russian):
http://lampes-et-tubes.info/rd/STS-6.pdfWhile atmospheric testing did result in global contamination, I find it very unlikely that clothing in running water would act as a filter and concentrate it to the levels mentioned in the reports, and do so to such an extent that beta but not gamma would be countable.
Beta-only radiation also firmly excludes theories of contamination from the abandoned mining camp. Any minerals containing uranium are going to be alpha, beta, and gamma emitters. An alpha scintillator, or simply using a mica pancake probe with and without a paper shield, would easily detect the abundant alphas emitted by uranium. This also rules out thorium contamination from a lantern mantle; that's also an alpha emitter.
I've also heard people suggest that Krivonischenko's work at Mayak, or Kolevatov's work at a secret institute in Moscow, could have resulted in them having contaminated clothing. That seems unlikely, seeing as:
1. Nuclear facilities, even in the USSR, go out of their way to avoid contaminating worker clothing. You typically change out of your street clothing and into clean work clothing at the start of your shift, and strip out of it and shower at the end, precisely so you don't contaminate clothing you wear home.
2. Even if there were a failure, and one of them "took their work home with them", is it likely that the same contaminated work clothing would then be worn on an expedition in the Urals?
3. Even if that were the case, all four of the hikers found in the ravine had varying levels of contamination on their clothing, which I can't explain via this mechanism.
It seems most likely to me that some time after all the hikers had assembled together, and before the bodies were found, those four (at a minimum) were contaminated with a beta-only emitter.