I don't think that such a miniaturized low pass filter is something that can be done at home
I would not reach for active filters or IC passives here unless you had to have something that worked every time, across all manufacturers and devices.
The thing is, @jaclaz's ferrite bead idea is a DIYable passive filter, one you can try at home. Moreover, it mimics actual problems you can see in the field, being too much inductance on the wire for the parasitic capacitance present. Kinking the wire and unkinking it creates a tiny parasitic inductance. It's a real effect your techs will deal with. All we're doing here is trying to figure out a way to get the effect reliably so the lesson doesn't fail due to clever electronics in the PHY compensating for the purposeful nonidealities in the hacked-up cable.
Simplified, equation (1) from my link above is L=X/(2πf). It gives you the inductance of the bead (or beads, in series) needed to get the effect you want. 2 and π are constants, and f we know from the 10Base-T specs, leaving the parasitic capacitance of the cable as our only mystery variable. I can't measure your cable for you, but I'd expect something on the order of 50-100pF per meter. (I could measure one here, but I'm lazy, and it wouldn't be
your cable besides.)
According to Wikipedia, 10Base-T dates to the days where they used a simple MHz-per-Mbit encoding scheme, so f=10MHz. An
online calculator I found gives X=160Ω, which means L needs to be somewhere around 2.5µH to get the desired corner frequency, presuming my 100pF-per-meter value is correct.
More inductance (more beads, more kinks and coils in the wire) pushes f down, so if one 2.5µH bead doesn't do it, try two in series, three, whatever it takes.
Also, don't neglect the "turns" through the beads, as that increases inductance. There are beads meant for passing the wire through multiple times. This adds bulk, but you might be able to stagger them down the length of the cable well enough without utterly wrecking the signal.