I switched the bulbs and still only the right one triggers the warning.
I forgot to test without any bulbs, but it seems its the detection thats different or defective, not the bulbs.
Put both original bulbs back and check that the warnings have reset without further interaction required anywhere.
Then, remove only the left one. See if the warning appears. If not, then obviously that side has no warning capability (which would be surprising). If the warning appears, then the issue is due to borderline differences within the internal impedence of the real bulb versus your replacement bulb which can be solved as described below.
Either way, whether bulb-blown warnings exist or not, you need to place an parallel resister to remove some impedence from the new bulb assembly. Some LED replacement bulbs have built-in resistors to prevent this problem so that its characteristics are so similar to filament bulbs that such warning systems do not get triggered.
Do the original bulbs have a Wattage (25W?) and Voltage (12v?) level markings on them?
What are the Wattage and Voltage level markings on the replacement bulbs?
With this information, the parallel impedence can be calculated and you need to put such a resister in parallel with the bulb. This will take some current around the bulb.
To explain it using a water analogy, imagine you have a water stream passing through your village. You then have a watermill where the water wheel spins with the flow of water resulting in some benefit such as milling the corn or whatever. The water flow is your current, the water wheel is your bulb, and the cornmilling is the light produced by the bulb.
Now you replace the water wheel with a different spec of water wheel that spins with greater resistance in its mechanics. The water flow will still make the wheel turn, but at a lower speed. As a result, the water flow is slowed down and starts to backup further upstream.
The bulb blown warning is detecting the slow-down in water flow and trying to warn against the risk of flood upstream.
The solution is to make a channel just before the water wheel to take some of the water around the water wheel and rejoin the stream after the water wheel. As a result, the flow of water continues and the warnings of water flow (bulb blown) disappear. If the channel is too small, it will not divert enough of the water and will be inadequate. If the channel is too large, it will divert too much water and starve the water wheel.
The size of the channel is the resister that needs to be placed in parallel around the bulb to draw some of the current flow and thereby cause the total current flow to be enough to not trigger the alarm.
