It actually sounds like quite a few things could be going on.
Firstly I'll address the uvb. Right off the bat, coil bulbs are known to cause eye issues as you mentioned, and their output of uvb is extremely limited. You will want to replace this as soon as you can with a long tube
uvb light, preferably a reptisun 10.0 t5 bulb, but since you have a t8 fixture, start with a 10.0 T8. Tropical bulbs are meant for shade dwellers, chameleons etc. Dragons need full desert like sun exposure. Without adequate uvb, they will suffer health problems and one's already showing signs of health issues.
<<< I've not heard or read of any problems with using your brand of UVB coil.
I know Arcadia and ExoTerra make very good (safe) coils. Been using UVB150s and UVB200s for near 8 years for skinks and dragons and no issues seen that are attibuteable to these compacts, only time I had an issue was when I forgot to replace an older UVB200 (it had decayed and was no longer producing sufficient UVA and Rex got the shivers for a few weeks , a new UVB200 fixed that).
The big problem is shop brand compact UVB bulbs like Zilla, ReptileOne, and others who rebox and rebrand cheap Chinese clones , these have substandard (different cheaper) phosphors in them and produce a very poor spectrum of UV which will harm the eyes and health of reptiles exposed to them, and also have very poor quality control in manufacture.
old but relevant viewtopic.php?f=34&t=112482
EXTRACT AND NUB OF THE PROBLEM WITH CHEAP NASTY CHINESE CLONES (TUBES AND COMPACTS)
There seem to be two basic problems with most UVB lamps (both tubes and compact lamps) manufactured cheaply in China.
Firstly, they appear to use a type of quartz glass that allows UVC and very short-wavelength UVB through it (unlike European glass, which I believe is very low-iron borosilicate) and also, when extruded into tubes, creates linear defects, like microscopic spurs inside the tube, which don't get covered evenly by the phosphor. There are therefore thin streaks of glass with no phosphor - you can just see them when you look at these tubes - and UVC can leak out through these streaks.
Secondly some of the phosphor blends they use, produce un-naturally short-wavelength UVB. This too escapes through the quartz glass.
The result is a lamp which can cause eye problems. It's not just cheaply-made Chinese compact lamps which can cause this. I have tested several brands of linear tubes with identical problems.
Unfortunately , money hungry companies still sell these all over the world and promote them as "JUST AS GOOD" as the more expensive higher quality brands, and lots of reptile keepers buy them as parts of kits marketed as being PERFECT for beardied dragons (in our case), ie ZIlla.
I highly recommend moving the T8 reflector hood UNDER THE MESH TOP ,as the mesh will be blocking a large amount of the UVA and UVB the tube produces , and install a good T8 10%UVB or higher rated tube , as bearded dragons need extreme levels of UVA and UVB to grow and thrive.
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being a bit closer to where the beardies are will be beneficial as they will see a higher flux of UVB which will help them better metabolise their dietary calcium to VitD and will help them grow and thrive.
I also recommend if you have light cycle about 12 hours per day to bump this up to 15 - 16 hours per day as this will be more like the daylight hours beardies experience in their natural range in summer.
Secondly, and I know you won't want to hear this, but you'll need to separate the dragons asap. Your little weak one is likely being dominated by the other, resulting in stress, less exposure to lights and basking, and likely less food. Dragons are not social creatures though they are often showcased housed with other babies, but things can get bad very quickly. Please invest in a second setup if you plan to keep both and spare them both the stress of trying to coexist in one enclosure.
<<<< I TOTALLY AGREE - they need to be separated at once before things go aear shaped .
You've been lucky to have no domination problems going on at 4 months, I think my Rex was starting to boss Puff around from about 3 - 4 months , and I was definitely seeing a difference in growth rates, Rex was getting a fair bit heavier and longer (both started off pretty much the same size at Rex (7g) and Puff (8g) at 6 weeks old).
So I brought my second 100L plastic tub (used them as rearing tubs for my tow Bluetongue Skinks as babies and juveniles) back out off my shed where it was in storage , and recommissioned it as her new rearing tub ( change of UVB from 13W UVB150 to 13W UVB200 , and new higher wattage mini spotty and it was good to go), and evicted Rex into the new tub. No more problems and Puff eventually caught up with his growth to where Rex was and wound up about 15g heavier when full grown at 2 yr old.
Currently have my two juvenile Peppa and Toothless in the same rearing tubs. .... sure got my money's worth out them as I've used them to raise 6 lizards from baby to subadulthood ( 4 beardies, 2 bluetongues ) and they've been in virtual continuous use for over 7 years.
PM me if you want to know how I set up my 100L rearing tubs. (I think plastic stackable storage tubs with clipon lids and wheels are fantastic rearing tubs with a few minor mods , and I'd select tubs 80L - 140L as your rearing tubs).
The other option is to partition your existing tank into two equal halves, sharing the same T5 or t8 10% UVB (MINIMUM) tube in a reflector hood and if you are cleavour sharing the same basking globe (over the partition.
Are you able to take photos of your dragons and the setup?
https://www.beardeddragon.org/useruploads