I think you could either center tap the input transformer on the loop side (prefered method), or add two 1K Ohm value resistors (or similar) from either side of the loop / transformer connection to ground.
My guess is that you have inadvertently scraped some of the enamel insulation off the transformer winding(s) at some point, as it's very easy to do this.
I use Kynar insulated wire (the type sold for wire wrapping) wherever possible, as this is much more resilient. However this doesn't work with very small cores, as the wire is too thick, so in this case I just have to be extra careful when pulling the enamelled wire through the holes in the core.
CAT 5 & 6 has a characteristic impedance of 100 Ohms per twisted pair. You can build an impedance matching balun or use two pairs connected in parallel and a 1:1 balun. However if you use this method beware, as some CAT cables use different numbers of 'twists per inch' on the different cable pairs in order to reduce cross talk between pairs. This results in some pairs being electrically 'longer' than others., which may cause problems if you connect them in parallel to halve the impedance.
I've now spent a bit more time playing with this application.
It's not the easiest thing to setup or initially use, but it does have some quite impressive features such as the ability to import various online databases and parse .xlsx files such as the 'Numbers and Oddities' and 'ITU monitoring' databases into a set of local databases that can be searched, sorted and indexed.
When driving a local radio or the Twente WEB SDR (or others of that type) it's very handy for click tuning, ID'ing or scheduling stations.
Definitely worth spending a bit of time with to get the hang of it, plus it incorporates some really good ideas and useful features.
The split is to avoid a continuous 'screen' (Faraday screen to minimise E-Field pickup) around the loop. But if one end of the center conductor is connected to ground you may have a problem.
Check signal levels with the loop normal and 180 degrees as suggested previously.
A 1:1 transformer on the loop input may help improve balance. Try six bifilar turns through a BN73-202 core as a starting point. One winding as the (balanced) loop input (primary) the other as the output (secondary) unbalanced connected to the amplifier input, with one end of the winding connected to ground. This will also provide galvanic isolation.
You can also try a few more turns on the secondary as an experiment as this may improve reception on the LF bands.
If you made a balanced loop amplifier with a much lower value of input impedance (LZ1AQ or Wellbrook copy) you wouldn't need the loop outer screen to minimise E-Field pickup.