There would appear to be an irresolvable conflict in that episode 3.08 must occur before August 11, yet episode 3.07 must occur after September 3. Alfred talks to Ivy about seeing the film Way Down East , which was an American film and released in the US on September 3, 1920. I expect it was a bit later than that before it found its way to Yorkshire.

Otherwise there are not many clues on dates. The best clues are in the clothing since we already know the foliage cannot be trusted. Multiple characters are seen in episodes 3x04 through 3x07 wearing overcoats and winter style gloves, so it is either still spring, or all the way round to fall. The reference to Drumgoole/Drumgool Castle is entirely fictitious, so cannot be used to date anything.

Tennessee about to ratify the 19th amendment is a complete red herring. At least one state in the "solid South" needed to break ranks and no one knew which state it would be, much less that it would be Tennessee until it passed by a single vote.

I would say that it is much more likely that Fellowes got the date of Way Down East wrong, than the dates of the collapse of the Ponzi scheme and the ratification of the 19th amendment. Thus I would think episodes 3x04 through 3x07 occurred in the spring of 1920, but hard to pin it down any closer than that. 

So, my point stands that it was June/July for baby Sybbie?

If we throw out the release date of Way Down East, then based on clothing episode 3x07 does not look like it could be much later than April. Episode 3x08 would then jump ahead to about August. Episodes 3x05 and 3x06 are clearly only a couple days apart. Sybil does not appear much in 3x04, and when she does is mostly wearing heavy clothing, so it is hard to guage how far along she is in that episode. 

It does seem to imply, however that there must be at least a month or two between 3x04 and 3x05. Robert would have been even more furious if she traveled from Ireland close to her due date.

He'd have been furious yes, but he'd have been EVEN MORE furious if she had given birth alone, but you are probably right.

I don't know whether it's more adorable or disturbing that you're using DA as your source for the minimum possible interval before a woman would know she's pregnant.  Cora, a perimenopausal woman who, as Dr Clarkson tells us, was cycling irregularly, plausibly might not have suspected pregnancy until the fourth month... or might have dismissed it as an explanation for her symptoms because of her age.  It's highly unlikely that Sybil, a newly married woman of prime reproductive age, and notably, a nurse, could have gone four months without knowing she was preggers.  It's a safe assumption she would be well aware by the two month mark, presumably having missed two periods and probably suffering morning sickness (which is often severe with preeclampsia).  Whether she would have written home that early is another question.  Three months was the traditional period to wait before announcing a pregnancy, because the rate of miscarriage was so high during the first trimester.  Would she have made an exception for her mum?  

Anyway, I have no issues with your inferred timeline; just the biological assumptions on which you based it.

Do remember, back in 1920 they did NOT have pregnancy tests back then, and since she - a NURSE - missed eclampsia, she could have missed the signs, especially in a war torn Ireland.