Any amplifier should pass what I call the "barrel-connector test". Placing it in line with the radio should not increase the SWR seen by the radio any more than a barrel connector and coax jumper would do.
For a long time now, that amplifier has been sold using a legal loophole. It's configured as a CW transmitter by installing a small circuit board that has an oscillator circuit with an empty space for a quartz crystal. A tiny 2.5mm jack on the rear panel is for your morse-code key.
This is a fig leaf to cover the private parts. Nobody with more than a gram of sense uses it the way it's set up from the factory. At least 999 out of 1000 of these sold will get "converted" back to being a non-legal linear by removing the parts that turned it into a pretend CW transmitter.
Here is an instruction sheet for that procedure:
http://www.cbtricks.com/Amp/txstar/dx400v_dx500v/dx400v-500v_mod.htm
My strongest suspicion is that the conversion was botched, and something is connected to the amplifier's standby circuit where it should not be.
Here's the schematic of the current "DX500V" model. Bear in mind this amplifier has been produced in various versions for around 30 years or more. Yours may be a bit different, depending on the age.
http://www.cbtricks.com/Amp/txstar/dx400v_dx500v/graphics/ts_dx400v_dx500v_sch.pdf
Most amplifiers like this one use two relays. One of them transfers the antenna and radio when you key the mike. The other relay takes the receiver preamp into our out of line.
Texas Star chose a different setup. It has only one relay. As a result, the path your radio's barefoot transmit signal takes is a little convoluted. First passes through C3, a 1000pf disc capacitor and then into the input side of the relay. From there, the receive-side contact sends it into the preamp switch, back out (if the preamp is off) and then to the output side of the relay. It then comes out of the relay's output-side contact pin and into C44 a .01uf disc capacitor and from there to the antenna socket.
There is a small RF choke coil L1 connected to the antenna side of the relay. Your radio's barefoot transmit signal should ignore this choke coil, as if it were not there. If this coil was overheated the varnish that insulates the wires on it will burn away. This causes the separate turns of wire to short to each other. A damaged L1 will cause the kind of trouble you're seeing.
But that's the list of things that can be blamed if placing the amplifier in line with the radio causes the radio's SWR to rise.
Gotta be one (or more) of those.
73