Sorry my reply is a little late, I rarely check the forums now-a-days.
Your problem is rather simple, if I'm seeing it correctly.
you should have .png after the weapon's name in the anim sheet like this:
<animSheet name="laser_burst_6" w="780" h="125" fw="51" fh="124">weapons/laser_burst_6
.png</animSheet>
<weaponAnim name="laser_burst_6">
<sheet>laser_burst_6</sheet>
<desc length="15" x="0" y="0"/>
<chargedFrame>9</chargedFrame>
<fireFrame>10</fireFrame>
<firePoint x="32" y="3" charge=""/>
<mountPoint x="0" y="72"/>
</weaponAnim>
Hopefully that should fix it.
Its rather hard to respond/co-ordinate on the forums, so if you have any future questions and want a response in something quicker than a 5 day timespan, I recommend joining the FTL discord, where you can contact me at Puffias Leroy McBillington III#1234
https://discord.gg/btqm3cY