Sure you can do this without removing the dash,....this is not a Volvo 242
The large glassfibre box under the gloves compartment comes out fairly easily, after you disconnect the heat-control wires, the water in/out, and a few other things that you'll find as you go along.
Do NOT be tempted to install a thinner core from some japanese car,......they will not give sufficient air temperature when the fan is on full power. I also took the oppertunety to put new "fabric" on the hinged "doors" inside the glassfibre box (The "doors" that regulate where the air goes). Maybe an idea to order this "fabric" when ordering the new core?
I also used sealing compound then mating the two 1/2's of the glassfibre box together.....make a "drain" to allow future leaks to be detected.....
To avoid the risk of an air-pocket forming inside the new core(reducing the effect of you heater),.. connect it the correct way..(water in/out). Good luck to you !