I think this is a good opportunity to also formalize the notion of “fill the tubes with X”
That’s good thinking, and a good way of expressing things so people can ask a simple question, look up a simple mechanical property, and get their answer.
I think the relatively poor results show why in a situation where strength is massively unimportant relative to stiffness, the practical solution is to augment diameter. This is what’s motivating me to look for things I can stick on the OD.
(That and it’s a heck of a lot easier to prep the tube’s surface on the outside than it is on the inside.)
I ran the numbers for Graphlite, a pulltruded CF rod which is one of the best in the industry. Their solid rectangular rod is 4.32mm wide and 1.53 tall. It has listed MoE of 142MPa.
Three graphlite rod’s moment of inertia is 1.91e-9, and assuming 1"x0.049" tube this gives
\psi = \frac{1.91e^{-9}*142e^6}{6.91e^{-9}*200e^6} = 19\%
So that’s in the ballpark of any ID solution, and uses a lot less material.
On the downside, I’m assuming the rod is held on with thin VHB, which while it is an awesomely good adhesive tape, its thickness and low shear modulus will certainly somewhat reduce the rod’s stiffening effect.
And of course, bumping up the diameter to 1" EMT is both cheaper and more effective, yielding a 45% stiffness gain for only 3% more weight vs. a 1"x0.065" tube. I think the conclusion is that stiffening of any form-- OD or ID-- will only make sense if there is no alternative.
See Modular Lowrider build (can use any steel tubing from 18mm (1/2" EMT) to 32mm (1.25")) for the alternative. 
P.S., it’s still really cool to me to think that a length of packing tape can have a measurable difference on a steel tube!