Tektronix support won't be of help as the scope is looonnnggg out of production.
Here are schematics for the TDS544, which will be the same for the front panel (and mosrly the same for the other parts) in your unit.
https://w140.com/tekwiki/wiki/TDS544AAs mentioned before, you don't need a specific probe to see the calibration signal, in fact during the official adjustment process a BNC plug with alligator clips on fly leads is used to measure the signal, it's as simple as you can get.
I also vote +1 on replacing ALL the SMD capacitors. I have a TDS500 scope on the bench at the moment and the caps on the front panel had all leaked, requiring replacement of some resistors and transistors along with the caps (luckily the capacitors on the processor and acquisition boards weren't leakink, they are being replaced anyway), so I would be checking that first up.
Also, while you have it apart, replace C17 on the PSU circuitboard with a high ripple current rated, low-ESR cap. It's a 47uF, 80 Volt part, but you can go higher in voltage rating for some overhead and better specs.
This is known for going high ESR and it ends up taking out some TVS diodes and a power transistor for the standby power supply section. You'll know you need to replace them all if your scope refuses to power on one day..
Here we go, I found some notes I put up on the tekscopes groups.io page:
Replace C17 (47uf) in the power supply if present (There are two versions of the PSU, C17 is on the version with the metal box that fully encloses the mains input plug/fuse area).
It is critical for the standby power and as such works hard over the years as it is always in use whenever the unit is connected to mains.
When it wears out and the ESR rises, it puts stress on some TVS diodes that have to take spikes and junk that the cap should be filtering. They then die and take out a BU508A transistor that ends up killing your standby power and then your scope goes dark.
The cap is a low-ESR Nichicon (should be about 0.39ohms), and I've measured them at up to 2 ohms, so out with it and in with a new low-ESR cap.
Also, check for the four 1000pF Rifa caps in the PSU too...
On the front panel board, there are two 10V caps (C22 and C23) that are subject to 15V, they should be replaced accordingly too.