Since Nickel alloys do not solder , yet may oxidize with heat, you must use a good gas-tight mechanical compression connector, such as the rivots that you found or an ultrasonic welder. ($$). a.k.a. Cold welder.
There is a lot of physics design and bread chemistry that goes into a good toaster design.
First of all to create all the chemical reactions on toast, you need a surface temperature of 150’C for the Maillard Reaction to occur, with a max of 10’C tolerance at the bread surface, otherwise you just have warm bread or burnt toast.
To achieve this you need a consistent thermal resistance from the red filament of around 760’C and enough gap between the filament and the mica and the filament and the bread to create a consistent temperature rise of the bread surface and not shadow the filament from convection air around corners causing to rise to orange. Remember the Arhennius Law? every 10’C rise reduces the failure time by 50%. This why light bulbs last 1500 h or even fail in months when the elevated temperature of a small glass enclosure fails to release heat.
Soldering above 1000 ‘C might be possible, yet accelerate the failure rate of the wire creating a weak link.
So if I were considering a design, I would examine the designs of 1940’s flipflop toasters that last decades longer than the modern fancy looking ones of inferior design ( for Nichrome<>mica details that affect reliability, radiant power, & thermal resistance with NiCr attachment only)
- Hotspots in filaments and weak link effects. (Arhennius Law)
I’ve tossed many a high quality brands of toasters with fancy exteriors that failed to last as long or work as quick as what I remember using over half a century ago.
So get a rivet gun and be done with it.
Then examine the temperature variation of the filament and the bread surface for various thicknesses to improve the temperature. Even better , add a sensor to regulate the surface temperature with PWM or a Triac using a well placed $2 RTD and IC to sense temp., using the Maxim 38165 IC