OK thanks Dan for the excellent pics and big is good, I'm old....
The relief at 0.007" you did not say if that is the treble side or the bass side. So what I would suggest is two things to start and I don't agree that we have eliminated the saddle, nut yes, saddle no I can't see from the pics for sure, for sure if there is not a flat spot or not enough break from the saddle on the forward curve away from the saddle top and I speak of the forward edge of the saddle top not the back edge which is in question too.
1) Capo in the first three fret area, your pick. Slack strings, pull pins and then examine the fit of each pin with a string in the pic some of the pins look high to me. When we have a new guitar we have to slot the bridge, top and plate for the strings. If the pins fit too tightly the string ball may not seat firmly on the bridge plate and this incomplete contact can rattle and buzz on a new instrument or an older instrument that the steward has upped the string gauge. So refit the pins the little band around them shown should not be in contact with the bridge but closer than I see in your pics hence the pins are not all the way seated and the string balls may not be in firm contact with the plate inside the box. The fact that you hear the noise on two strings suggests this is a process problem - the fitment of the pins perhaps.
2) Put the strings back in when you are sure they fit nicely and are seated nearly to the band around them. Pins should not need to be forced down they should easily fit completely. Tune to pitch, adjust the truss rod so that the side of the neck, treble or bass with the LEAST relief is very nearly straight. Take it to as little as 0.002" of relief and again I speak of the straighter side of the neck and this is always done tuned to pitch, truss rod adjusts. Measure action at the 12th for BOTH the treble side and the bass side. Remove and shim the saddle with any old thing including plastic cut from packaging material we just want to raise it for now and this is a temp shim. A shim 0.032" will raise action 1/64th inch. Shim the saddle to get the same action you had (good going on that by the way that is what I set a dread to with 13's in the shop for a flat picker with a moderate attack) to 5/64th treble side 7/64ths bass side.
What we have done here is straighten the neck as much as we can (I usually set my relief in this area anyway unless someone is ham handed heavy hitter) and that will also take any "body hump" which is a very common occurrence on early, small builder instruments, it will take the hump more out of play if it exists. A body hump is a rise in the fretboard at the body join and that is also where the strings lash out some of the furthest and a common place where if the action is too low for the amount of relief and if the hump is present there can be interference with the frets for vibrating strings.
3). One more thing to check for now and that is with the saddle removed draw a pencil line on the bottom and lightly sand the bottom on say your table saw table with 220 paper, something very flat until the line is gone. We want to make sure the saddle is flat on the bottom. Additionally the saddle should be able to be removed with minimal pulling and NOT tight in the slot. If it't tight in the slot it may not be completely seated and that can buzz too. On guitars with a pick-up under the saddle I like the saddle to be removable with only fingers but never, never rock back and forth. Without a pick-up my end nippers are used to remove the saddle but I don't want to have to pull very hard to do so.
So check saddle fit and how well the bottom of the saddle sits in the bottom of the saddle slot. Is this a commercially made bridge? Our shop might hit the bottom of the slot with our saddle mill to make sure that is flat too but you don't have this tool.
Do these things and see what results?
One last thing and that is shaping the top of a saddle.
I use the side of a pencil and color the entire top of my saddles and then I hit them with my favorite saddle file witch is a very old, pointed half round OOOO file. I file the saddle top until all the pencil is removed but there remains a thin pencil line not a thick line. A thick line is a flat spot a thin line is a peak which is what we want. We also of course want compensation so that thin line the High e is forward of the b, the G is forward of the d, a, low e, etc. Using a pencil you can eliminate any flat spot on the saddle top.
Dan sometimes we have to work on our hunches in this trade and what I have described we unaffectionately call "fishing with a net..." If it results in a fix and if you are unsure of what exactly was the offending thing that was corrected..... welcome to my world
Good luck and there is more when you get past this stuff.