Your problem is thermal.
The LEDs will only work if you don't overheat them. The "heat sinking" in these cheapie lights is only so good - after all, there's a way they got so cheap.
So you need to figure out the power actually going into those LEDs and insert no more than that - derating as necessary to not exceed the original overall power. The good news is that not pulsing will only improve efficiency.
The problem is, it's more difficult to surveil the power usage of a pulsed DC source, at least from the DC side. Perhaps you can wire a 0.1 ohm resistor in series with the power supply and measure voltage across the resistor, giving current/10. Also measure voltage and take samples at high enough frequency to see the whole pulse. On the AC side you can install a Kill-a-Watt energy monitor but that may not be 100% accurate - not least the efficiency of the power supply is difficult to know.
One thing you will discover is whether the LED itself is doing the shimmering. There's a type of LED called Chip On Board (COB) where silicon right on the LED emitter manages the LED brightness. If that is the source of the shimmer, you are out of luck.
Bare naked LEDs are current devices
Think about any random appliance. Toaster, USB charging block, water heater, whole house air conditioner outdoor unit, TV remote. All of them are given a constant voltage - 120VAC, 240VAC, or 3V - and they "magically self-regulate how much current they take*. The incandescent bulb does the same. **And people get concept-locked into thinking "all loads are like this". Not true at all.
It's funny because Edison tried thousands of filament designs, trying to find (by brute force) one that would self-regulate on constant-voltage. Constant-current wasn't an option because in 1890 that was only practical with AC power, and that would mean conceding to Tesla's AC system. Had Tesla been the master marketer instead of Edison, your light sockets would all be constant-current sources :)
Lighting in particular is not that way. All technologies of lighting (except incandescent) must be current-limited - take fluorescent. It takes a high voltage shot to initially strike the arc, and then, the lamp acts like a "dead short". It needs a ballast to limit lamp current to a survivable amount. Same with all the HID types.
LED is only a little more friendly. There is a voltage/current curve, but it's very steep. Drive it constant-voltage and you increase voltage a little, current increases a lot. Worse, that curve moves around based on temperature, age and binning. You can get away with it if you choose very conservative voltage, but then you won't get very much light. Indicators generally put a resistor in series with the LED to make it more linear, but in LED lighting you can't afford to add even more heat when heat management is already a challenge.
So LEDs for lighting are generally driven in a constant-current mode similar to a fluorescent or HID. That allows the driver to keep the LED right on spec in a wide variety of conditions and manufacturing variances.
This "constant-current" mode is actually a great way to dim them. Vary the current to set the dimming level. In fact, in this mode they are the finest dimmable light available, short of running an incandescent on DC.
I don't know why your LED is pulsing. Either they are using a cheap-mode power supply and picking up 60 Hz power supply ripple (120Hz if full-wave rectifying)... or they are intentionally using PWM as a method of dimming (which is fine). I suspect the former, as PWM is usually at too high a frequency to bother TV equipment, and 60/120 Hz is a close multiple of popular TV frame rates of 24, 29.97, 30 or 60 Hz/FPS.
That said, using "cheap Cheese from overseas" LEDs is a bad idea with TV production. The reason is the CRI (Color Rendering Index). It will tend to be either "all over the map" or "just bad", and typically poorly documented in any case. Hold out for 90+ CRI from reputable suppliers - it will make you look better.
