How to work tempered hardboard?

I've made some bins for my CD collection (and larger ones for DVDs) with drawer-like construction techniques. The front and back are one-inch (nominal) lumber, but for sidewalls and bottom, I've used tempered hardboard (like we used to call Masonite).
To hold the sidewalls to front and back, a rabbet and glue with some pin nails does the trick, though assembling it isn't a neat process. Front and back can take a dado to capture the bottom, but the sidewalls are only 1/8" thick, so to hold the bottom, I made tab-in-slot features with several 1" long slots in the sides, and with a dado blade cut into the bottoms (as a clamped-up stack) to form the tabs.
The problem: hardboard isn't what my tools are intended for. A 1/8" diameter carbide router bit, and a Rotozip-style side-cutting drill, make ragged slots (and it's not clear that the steel tool isn't overheating). A table saw leaves a furry edge (OK, that cleans up, by hand, with some sandpaper, but... it's a nuisance). I can apply a bit of shellac on the furry edges, then iron them flat with the old Proctor-Silex appliance, but one must be careful about the glue-surfaces.
Oddly, a brad-point drill makes a perfectly clean cut (to start the slot), so there's ONE tool that isn't a mismatch to this material.
My two questions: is there a way to form those tabs that doesn't turn so much material into sawdust (probably not, but I've gotta ask), and HOW can I make a clean slot in what is essentially just heavy paper? There's metal punch/die sets that could do it, but the supporting press tooling isn't gonna fit in the basement.
I can imagine a nibbler tool to form the tabs, but bandsaw (I've tried it) is slow compared to stacks fed to the dado blade in the table saw.
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whit3rd wrote:

Have you tried scoring the cut line with a knife first?
Really sharp tools will help a lot, but most edges are angled for longevity and not ease of slicing. I wonder if an ATB blade will help?
I usually cut with a saw and just deal with the edges.
Puckdropper
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On Monday, August 26, 2019 at 10:54:32 AM UTC-7, Puckdropper wrote:

It's usually a problem on the long-dimension rip on a 4x8 foot sheet, so the scoring would involve a modification to a table saw (possible, but messy) or some really long straightedge for guidance. I'm considering a narrow-kerf blade in a zero clearance throat plate (very thin material, otherwise I'm uncomfortable with narrow-kerf solutions).

Yes, a rip blade WOULD be a better (than combo) for this; currently, the plan is to do those cuts with a track saw, and I'm not sure I have the rip-blade option in my stack of small blades. It's worth looking into, thanks for the suggestion. Zero-clearance throat plate in the track saw?
Aerosol shellac on the edges, and a lick with a hot iron, is the fallback solution. For preference, don't try this with a good teflon surface iron: there's some abrasion.
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On 9/6/2019 12:07 PM, whit3rd wrote:

Has anyone tried using a Triple Chip Grind (TCG) blade for cutting this stuff? It works wonders on some materials.
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On Saturday, August 24, 2019 at 4:19:42 PM UTC-4, whit3rd wrote:

I find myself placing either masking or blue painters tape centered on both sides of the cut line...seems to reduce the rough edges a bit, especially with masonite door undercuts...
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