It depends on the architecture of the system you propose. At it's simplest, take the output of PlayIt Live into a mixer channel, return the mixer to PlayIt Live and stream the output.
Job done !
More sophisticated, have a computer "somewhere" streaming a schedule of pre-recorded material and take the "live" insert as a stream (if remote) or aux (if local).
To me "clock" tends to refer to music rotation. Are you thinking of hour segments in a schedule ?
I think I see how this is supposed to work.
You take a "core" or "sustaining" feed (perhaps as a web stream) for a proportion of the day (or hour) and make local material to insert at certain time in the hour or day.
You want PlayIt Live to run a master schedule that ensures that the core material and any promo, junction or interstitial material gets played and the local material from your mixer fills only the allocated slots.
I think this should work although it might not sound great at the junctions as the local mixer will (I think) just be cut and it requires the local material to run to exact time.
Your presenter will need to hear the output of PlayIt Live but not have it on the mixer (or at least not fade it up on a simple mixer) as otherwise it will all howl round as the mixer is a source in the PlayIt Live mix.
Hence any clips in the live material will need to come from something else.
Ideally you would want some sort of "clean feed" or "mix minus" from PlayIt Live.
There would be other possible approaches but I can't think of any other way where the schedule "master" is the playout (PlayIt Live) system.
Also, anything else I could suggest would cost many times what the simpler approach you have will.
If you can trust the presenters to respect the schedule then personally I would use the approach of
PlayIt Live > Mixer > Streaming software
Rocket Broadcaster (being just a streaming application) has more features than PlayIt Live's streaming plug in (particularly in the paid version).
I might also use a dedicated computer just for streaming.
The reason for the "might" is that for the part of the day that you use "full" automation I think you could use PlayIt Live as a source into Rocket Broadcaster and take the mixer out of circuit (I haven't actually ever tried this but I think it would work).
When you go "live"you fade up the local line source as "microphone" for Rocket Broadcaster and fade down "other applications" (which is where the PlayIt Live audio would appear during full automation).
PlayIt Live would also be a source on the mixer so can be used during the part of the day that you do a mix of live, sustaining and local recorded elements.
Without having tried it, I would look at creating clocks based on the elements of the "sustaining" feed you want to use and fixed time markers.
I would then use probably use a break note for the live elements and add jingles and promos either as files or from folders as needed.
There will be a fair amount of "finessing" of the clock needed to get it right.