I am in a similar position, but transfering in SIN on two separate tickets with QF and Silk Air. Last year I did the same trip and QF refused to check me through to Lombok from LHR but subequently discovered that the transfer desk at SIN is very efficient. I have emailed them and they say that I do not have to go through immigration and must simply transfer terminals and visit the transfer desk. They say that they will locate the luggage from the bar code and reroute it to LHR. It sounds too simple to be true, but maybe it's worth you emailing SIN airport to check?