QRH,
Check your Ops Manual but in every Ops Manual I have seen, indeed I wrote the FDP's for one operator, a split duty is an extension of duty, a continuous duty seperated by a break, and if a total (split) duty period is, by example, 13:21 then one needs that appropriate rest period before the next duty.
If your Ops Manual doesn't agree with this then your operator, or the individual that informed you that a split reduces an FDP, is an idiot.
It seems you are dictating, on the split, normal off/on duty (on blocks + 30 minutes followed by a 1 hour report), normally, if the operator has any sense, on a split these off/on duty times may be reduced by perhaps 50% on the basis, I guess, that you don't need to put the beast to bed and/or complete all end of duty paperwork and you are reporting back to an aircraft that hasn't been anywhere since you last left it.
But, a split duty is an extension of duty, does your Ops Manual, not the individual who informed you otherwise, agree with this?