Your sound engy is hacking the guitar's connection to the mixer. To what end, I can't say, but I can say that the change in cable is significant and is the primary reason for your woes.
Acoustic-electric guitars have a battery-powered preamp. To avoid that battery being constantly connected to the preamp, draining the battey when you're not using the guitar, the ground side of the power supply circuit is usually run into the jack using the "ring" contact of a tip-ring-sleeve or TRS jack in the instrument. Insert a standard two-conductor "tip-sleeve" 1/4 plug, and the ring and sleeve contacts are shorted together through the plug's sleeve, completing the battery circuit and powering up the preamp. This is why you're told not to leave your guitar plugged in overnight or when you're not using it.
The old tip-sleeve plug you were using worked for this purpose, and it gave the mixer a usable if probably noisy and rather mid-heavy signal. But, when the engy changed it out to a TRS-to-XLR cable, two things happened.
Your guitar's preamp can no longer complete the power circuit to ground, because the "ring" and "sleeve" contacts only have continuity through the long cable run back to the mixer, where there's likely a 600-ohm resistor between those two conductors that provides an "impedance-balanced" circuit for microphones and other low-impedance audio sources. That reduces the current flow through the preamp below the minimum power requirements.
Your signal ground is likely wired to the sleeve conductor, which is expected for an unbalanced audio signal. However the mixer, given a three-conductor input, is expecting signal through the "hot" (tip) and "cold" (ring) conductors of the input, meanwhile the ground (sleeve) is considered pure noise and never enters the mixer's signal path.
The Epiphone player's guitar eventually worked because pulling out the plug slightly either used the sleeve contact as a jumper between the ring and sleeve to complete the preamp circuit, or if it was a traditional electric guitar with no preamp, the sleeve connector simply made contact between the ring of the plug and the ground contact of the jack without disconnecting or shorting the tip, to produce a valid audio circuit for the mixer.
The cheaper A/E probably worked because its jack design was slightly different, either connecting power through two contacts for the same conductor of the plug, or using a separate switch for power that isn't electrically continuous with the plug (this avoids the annoying "pop" you get when you plug in a preamp-powered instrument). Just because the guitar's cheaper doesn't necessarily mean corners were cut in the electrical design; there's a lot of other corners to cut in the construction of an acoustic instrument to hit a pricepoint.
Anyway, it's the "audio guy" who is screwing up here. The mistake wasn't swapping the cable, the mistake this whole time has been not using a DI box on-stage to produce the low-impedance balanced signal circuit the mixer expects from the high-impedance unbalanced signal circuit of the guitar. Your guitar should be using a typical two-conductor instrument cable, fed into a direct input box which uses a transformer to "isolate" the high-impedance circuit of your guitar from the low-impedance circuit output to the mixer, thus giving both devices the electrical circuit they were designed to use while transmitting the signal between the two circuits. Plug in a DI, or a tone-shaping preamp stage like a SansAmp or an L.R. Baggs PADI, and not only will you get your signal path back, the sound quality will likely be dramatically improved from what you'd been getting with just the TS-XLR cable.
The fact that other guitars "work" with the new setup is only weak evidence that something's wrong with yours, and in any case, the audio guy is there to make his system work with your guitar, so by telling you your guitar is the problem, when it's clear his change caused other problems for other people, the audio guy is just trying to dodge his responsibility to you as a fellow member of the worship team.
As for the status of your pickup and preamp, what's going to work depends on the sound you want, what parts of the existing system you want to keep, and how much you want to spend. I'll tell you that a replacement preamp and pickup, plus a knowledgeable luthier's labor time to swap out, can easily cost you more than the guitar's worth when you're done, and may exceed the cost of a new comparable replacement off the rack.
Replacing like-for-like, that Fishman Presys+ will set you back about $200, and that's if you're confident in your ability to swap out the old and new components so you can save the labor costs of installation. It's not rocket surgery, and it's a big help that the guitar's already got a system installed so you're not drilling/cutting holes, but it can still take you quite some time to fish wires and tighten down connections and fittings. If you're serious about what you do on Sundays, this might be an opportunity to upgrade the entire instrument, say to a Yamaha FGX series or possibly even one of the lower-end Taylors. The Fender will still work acoustically, so you can keep it around as a "beater", or you can trade it in to lower the cost of the replacement.
Either way, if you honestly suspect your electronics are going the way of "Total Incapability To Sustain Usual Performance", it's best to get ahead of that ASAP, before the guitar ends up crapping out on you in the middle of service.