I think that you are doing the right things along the right lines. At the moment, you reliance on one person to give the translations is leaving you open in that you rely on his word, as you said.
I say this because I have been in the same situation. A number of years ago I worked(had just taken the job in fact) as H&S bod for a large retail cleaning company whose main client says every little helps. Cleaner in a very small Metro style shop shop normally would finish the shop floor clean before shop opened. On this occasion one cleaner was off sick, so another had to come from a different shop after finishing there, meaning that in the second shop, she was cleaning when customers were on the shop floor. Floor too small for cleaning machine, so mop and bucket was the order of the day.
Cleaner was mopping the floor, elderly lady slips on water on floor, falls over, injured as a result. Cleaner continues to mop the floor around the injured person and does not understand the store manager's shouted repeated instruction to Stop, who eventually has to grab the mop from her to stop her.
Environmental Health officer got there on the next day before it was reported to me. Cleaner's training record was there, written in English, with the statement at the bottom saying: 'I have read and understood the above', followed by her signature. The cleaner understood yes, no thankyou in English, and very little else.
When I investigated I found that the area manager was the cleaner's son, who was born in Brazil but was brought up here, so spoke Portuguese and English, so he explained the training to his mum in Portuguese and got her to sign. But when the EH officer asked me - How do you know that the area manager didn't just say: 'Sign this mum, no reason to read this rubbish, just sign it and you can start' - I could not provide evidence that I knew that this was not the case and as a result was left in a difficult position. Company settled out of court and EH did not pursue on balance of probabilities, but is still cost the company about 6k for a broken ankle payout.
Third party corroboration was where we urgently went next in terms of veracity of translated documentation. Looks like you are going that way anyway, just thought that my experience could be beneficial to you. Certainly at the moment, you could look at what you would say to HSE/EH - whoever is relevant to you - in the event of accident/incident and subsequent investigation. Could you honestly say that today you know that you are sure that your workers have sufficient information, instruction and training? It will cost a bit to get that assurance in terms of another person/organisation confirming the translations but will save in the event of legal action. One question - are they just verbal translations of written instructions or are they written in the translated language? Otherwise how can you get a second opinion without a second translator coming to site?
Only time in my H&S experience that I felt very uncomfortable in the procedures in place. Best of luck.
Martin