To answer your questions...
If you get a skilled agent at LHR, they should be able to check bags through to MEL. It might be a little challenging for them though, so just bear with them.
You shouldn't have a problem with your second option. However, if you have a longer layover in CMB (compared to SIN) then ask bags to be checked in till CMB and re-check in there.
Originally Posted by
bdmaus
The T&C don't really say anything against it - they only seem bothered about using coupons out of sequence...
And even then it's okay provided:
3.4.1(c) a separate fare has been collected for the unflown sector(s)
Besides that - I have every intention of flying the sectors

Technically, I guess, you should be OK. However, EK has cancelled reservations on me in a similar situation to yours (I was holding 2 tickets - 1 in Y and 1 in J on an identical route - and was planning on refunding one of them) The reservation was reinstated but it was a headache.
Cheers,