How Low can you tune on a 35-inch scale bass?

35 is one fret shorter than 37. You can expect similar performance on a 35 bass in F, as a 37 in E with the same string, for example. I don't honestly feel that there are such extreme differences between the two lengths - afterall, would you say your open string sounds a ton clearer than your first fret, or that the instrument doens't intonate when capo'd at 1?
Certainly the extra length does help bring the required gauge down slightly and brighten things up if comparing in the same tuning. But it's a stretch to say 35" can't do tunings 37" can , like people do say. You could expect 35" in F# to be better than 37" in E, for example.
I would recommend a 190 gauge next to a 140 B, 105/80/60/45. It'll be pretty balanced in F# standard. The tone will suffer significantly going up to 200 and above to retain similar tension in drop E, so I'd just try with the 190 and see how it goes.
I can recommend Newtone strings (email them with your requirements and custom taper lengths for the F# and B), or Kalium if you don't mind a potentially long wait.

 

To answer the thread title question:

On 34"/35", the functional limit is far lower than you might expect. Garry Goodman of Octave 4 Plus has made some .400 strings for customers that are 'functional'. That is the largest gauge in existence i know of.
A .400 allows tuning to E00 (10Hz), 2 octaves below standard E.
You can continue to add mass and cause ever lower frequencies.

The actual limit is your own taste in tone and your tone quality requirements. As gauge increases, stiffness increases, causing a darker sound and increasing inharmonicity.

On 34"/35" with good strings, a good tone can be acheived down to around F#0 (a fourth below B).
As you go lower than that you have to be prepared to lower your standards of tone quality and accept a different type of sound.

There are many ERBassists who are happily using C#0 (2 fourths below B) on 34"/35".
Yves Carbonne uses a .266 tuned to B00 (octave below B) on a 34" scale.

So yes, 35" is enough for F#0 to E0.
For F#0 to E0, use a substantial gauge F#0 string, i suggest .175-.185.
'Kalium Music' are very good strings, probably the best for very low tunings, but be prepared for a possible lack of email response and up to a 2 month wait for the strings. The company is not malicious but is rather eccentric, uncommunicative and slow to deliver.

'Newtone Strings' make custom extreme strings and have a good reputation.

 

If you are choosing a bass guitar for this usage, make sure it has a modern top-loading and slot-loading bridge.
Through-body stringing might prevent the tapered section of the string sitting on the saddle, which is essential for very large gauges.
If the bridge forces you to thread the whole length of the string through a hole, that hole might be smaller than the gauge of the string.
With slot-loading you only need to pass the tapered section of the string through the slot, no bridge modifications needed and you are free to use even larger gauges in future.