No, this doesn’t sound like the HT is failing in the way that you mean. Usually when we talk about the HT failing, what is actually happening is the cancer has learnt to survive without testosterone. This is also referred to as being hormone independent or castrate resistant. If his testosterone is rising, then the problem isn’t that the cancer has become castrate resistant, it is that the injection has somehow not done its job in eradicating testosterone from his body. There are a number of possibilities - the last injection was given as a one month dose rather than the 3 month dose, or it had been stored incorrectly, or was from a faulty batch, or had been injected incorrectly.
That’s why a testosterone test would be so useful - if his PSA is rising and his testosterone is above castrate level, they may just change to another hormone. If his testosterone is below castrate level, they might add bicalutimide to disguise whatever testosterone is floating around or they may decide to add enzalutimide or abiraterone. I don’t think additional chemo will be top of the list right now.
As well as the testosterone test, I would be tempted to ask the GP practice to check the dose he was given for his most recent injection.