When speaking rapidly, it doesn't seem that I make a schwa at all when saying a phrase like, say, "the bus." It seems like I'm saying [ð̩.bʌs]. Is this a documented phenomenon?
It is well documented that a schwa can be very reduced, and significantly decreased in duration so that it is under 20 msc. The amplitude of the vowel is significantly greater than that of preceding [ð], which gives you a diagnostic for identifying a remnant of a vowel. The test would be to look an numerous tokens of "the bus" to see if the vowel is ever completely gone (while [ð] is maintained). Of course you can easily get a trained linguist to say [ðbʌs], so the question has be be something like "does it naturally happen in speech?", and not "is it possible to pronounce it this way?".
You might be able to set up contrasts if you are willing to embed the phonetic frame of interest in something else – namely, ...VðəCV... versus ...VðCV..., for example "bathe Robin" versus "pay the robber" (one would spend more time cooking up suitable pairs). This might allow you to answer the question "does the vowel ə ever go away completely?". However, it cannot tell you whether [ð] is syllabic, because that isn't a measurable property. That is, only fairly abstract phonological analysis can tell you whether you have [ðbʌs] versus [ð̩bʌs], and there aren't any facts of English phonology what would make that decision for you.