The SI-11BG is a tube designed for high radiation detection. You will typically find it besides a more sensitive tube in soviet era military radiation screening gear.
The LND-712 tube is very good and highly sensitive, also capable of detecting alpha radiation. But since it's a mica windowed tube, it's mechanically quite delicate.
A good, rugged and fairly sensitive tube is the russian/soviet SBM-20. It will detect beta and gamma (x-ray) radiation and it's widely used in contemporary, commercial radiation detection gear. These tubes are easy to find and quite inexpensive.
If you want a really sensitive tube, the SBT-10A may be a good choice for you. It's characterized as a highly sensitive (low energy) beta and gamma detector by the manufacurer, but it will also have some sensitivity for alpha radiation, though you've got to get really close to the source. Due to its huge mica window, it needs to be handled with care. I once designed a geiger counter around one of these tubes and utilized their cellular design with individual anodes to provide a uni-dimensional spacial resolution which proved more tricky than anticipated, but finally worked.
Here on the geman language "Geigerzählerforum", I reported on that build... (you may use an automatic translator if required).
To increase the sensitivity of your instrument, you may always parallel several identical tubes. Use individual anode resistors in this case. It may also be beneficial to use capacitive signal coupling at the anode instead of the more traditional cathode-coupled current amplifier. Especially on large tubes, the massive cathode may pick up all kind of noise and cause signal interference.