I’m in the same boat, in terms of girth. Over the past seven years, which is when I first got a stroking device (first tried a Vorze A10 Cyclone, but it was simply too small to even fit), I’ve evolved how I handle things.
First, avoid being trapped into an ecosystem. The Launch is the first stroking device I had myself. I also have a Handy and an OSR2. The latter is what I actually use these days.
Quite early on, I took an extra Fleshlight holder and made the modification you can see in this image. Basically, cut along the red lines (continue them as appropriate around the entire circumference), then use heat to bend the arms straight at two separate joints. Using the holder in that picture, I can put a whole host of different sleeves, ranging in size from large ones that require removing the inner ring to fit, and small ones that require using a cut up old Fleshlight sleeve to fill the space between the ring and the sleeve (e.g. Tomax Lilith toys). I use multiple rubber bands to secure the sleeves inside.
That opens up your options for sleeves.
Next is the issue of lubrication. I went the route of a manual lube pump, which is just a syringe containing a lubricant with a tube attached. The tube is connected to the same Luer lock connector as on the syringe, which I created adapters for (which must be 3D printed). One is suited for Fleshlight sleeves. You just insert it into the opening at the end, and use a rubber band to secure it in place around the narrow waist. The Luer lock goes into the top, and the tube connects to that. When you depress the syringe a bit, lube is forced through the tube and into the top of the sleeve. The other is used, in conjunction with a threaded cap, with other sleeves. If the sleeve is not already open-ended, which is the case with everything I use, then you need to create a new, narrow hole to pass the lube adapter through. I secure the adapter laterally to the arms of the modified Fleshlight case with rubber bands, to keep it steady and provide some resistance to the sleeve sliding forward on the backstroke. If you want the STL’s for those adapters, just ask.
If you stick with Fleshlight sleeves, then you’re basically stuck with the same tightness and material softness. Back when I first got one in 2001, the material was a lot softer than it is today. Putting in the effort to allow using other sleeve is worth it to have more control over what you use. My current preference is sleeves made by Tomax, which are sold as “very soft” or “rich soft” (the latter being the softest). The texture of the sleeve, as well as the stiffness of the material, will affect how well it holds on to lubricant. As does the viscosity of the lube you’re using. But none of the sleeves entirely avoid the issue you’re facing, which is why it’s worth rigging up some way to dispense lube without interrupting your session.