Those bidirectional translators are designed to work with logic chips. Below is a typical schematics (taken from Sparkfun). As you can see, they work similar to open drain circuits, i.e. the outputs are pulled high by relatively large resistors, and only actively driven low.
Your G3MB relay has only 440 Ω input impedance and they require at least 10 mA LED current to switch on. 10k pull-up on level translator can source only about 0.5 mA.
So, get rid of that 4-line bidirectional translator. You only need one signal and in one direction only. Use simple high-side switch to control SSR.
N-Channel M1 should have logic level gate voltage (something under 2.5V), P-channel M2 should be rated at least for 30 mA.
simulate this circuit – Schematic created using CircuitLab
Note that this only works if your LGND and HGND are the same common GND. The translator that you use has them connected internally. You did not specify how you convert 5V to 3V. If it is LDO or Buck then your grounds should be connected already, if it is isolated converter you need to connect them manually.
There are many alternative solutions, for example optocoupler here. You might even be able to reuse that level translator of yours if you connect its output to the gate of M2 in the schematics above. In this case the translator will replace both resistors and M1. The control signals on GPIO2 should be reversed though, with 0 being "on".