That was a fun challenge! I'll go ahead and share my observation.
After Scott challenged me to hold Sirius for as long as I could after sunrise while I was "nearby" in Arkansas, I set my alarm for 7am the next day (Oct 22nd) and kept my eye on Sirius. The transparency had been excellent the night before, but that morning there were cirrus clouds over Sirius which thickened as time went by, and I think I would have been able to get a longer time if there had been no cirrus clouds over it. Still, I was quite happy with the time I got!
I recorded many descriptions of the conditions and visibility of Sirius periodically, using my pocket voice recorder, which I record all of my observations on. Because I wasn't sure how accurate the voice recorder's timestamps would be, I had time.gov open on my phone (and later my mom's phone, after mine died), and read out the precise time on many of the voice logs. I was able to determine that my voice recorder is slow by just about exactly 33 seconds.
Local sunrise was at ~07:26am according to the NOAO computational definition, which is what I use to keep my observations standard. (It is, however, interesting to note that the apparent sunrise would have been noticeably earlier due to observing from a mountain at an altitude of 2180ft above sea level: SkyTools estimates apparent sunrise would have been 07:21am.) My last record of "a couple of good pops" has them occurring at about 8:13am, before I lost it completely in the clouds. Therefore, I was able to see Sirius for 47 minutes after sunrise!
Here's a timeline of my observation that morning:
-16 min - 7:10 - easily visible, I immediately abandoned my "post" to go watch the sunrise.
-8 min - 7:18 - Still easy with direct vision, light patchy clouds begin to form, I leave again to see if I can spot the fog in the valley from my vantage point.
-5 min - 7:21 - Still easily visible, I dash inside for more cold gear.
-3 min - 7:23 - Still holdable with direct vision. Gauzy clouds now covering Sirius.
Sunrise - 7:26 - Still 100% holdable with direct vision, but with "little, insignificant notches" where it rarely seems to flicker instantaneously.
3 min - 7:29 - Getting faint, and starting to need to know where to look. I lose it when I move my eye off of it, but when I have my eye on it, it's 100% of the time with direct vision.
6 min - 7:32 - Seeing appears bad, and Sirius is now 90% of the time with direct vision, as staring too long causes it to go away.
8 min - 7:34 - The cirrus clouds passing in front of it are thickening, but so far it has remained visible through them.
8.5 min - (30 seconds later) - Sirius starts to disappear behind the thicker sections of the clouds.
11 min - 7:37 - I can now see the sun if I turn around. Note to self: Don't turn around!
12 min - 7:38 - I keep losing Sirius behind the still-thickening cirrus, but when it appears in a gap, it's still mostly visible with direct vision.
15 min - 7:41 - Easily visible with direct vision 100% of the time when a cloudless patch appears.
18 min - 7:44 - Sirius is holdable with direct vision, however, I have to be careful to keep my eyes directly on target.
22 min - 7:48 - Sirius is now difficult with direct vision, and with slightly averted vision, it's pretty much holdable.
31 min - 7:57 - Cirrus clouds are now causing long gaps in visibility, but when visible, while it's not easy, it's not at the edge of visibility yet.
33 min - 7:59 - 50%-40% of the time with averted vision, when viewed through a relative gap in the clouds.
36 min - 8:02 - Still got it, but man, it's getting hard! Starting to look more ghostly, like a phosphene or visual artifact, but 95% sure it's not in my vision, as it follows the sky when I move.
38 min - 8:04 - Very slight, and getting to the point where the pops have to be very long in order for me to be sure I'm seeing something. "At that ghostly threshold now." Cirrus clouds nearly constant.
47 min - 8:13 - Only a few pops since 8:04, and the last few good pops were at this time. Cirrus clouds remain thick.
Here's a photo taken in the direction of Sirius around 8:00am, showing those infernal cirrus clouds:
I've added an article about my observation to my website, complete with the original voice logs and full transcript, if you'd like to check it out: