Real cars have hydraulics
Silver solder is WWWAAAAaaaaayyyyy to hot. Isn't it's melting point about 450degrees versus about 200 for normal plumbers solder you would have melted the solder in half the radiator before the silver solder was workable. I haven't used silver solder myself, it would be handy if you needed joins with strength (then again I'd probably use a welder in that case). You also need to use flux, the solder will flow where ever the metal is shiny and the flux has flowed.Be carful not heat too much as all of top solder might melt and destroy it. Mine had same problem and I tried to blow torch silver solder but I Now I learned, normal solder should have been used. So mine was destroyed . I took mine to PRO and bill was over $400.00. Now I am scare of leaking radiator. Man put new copper core , he said copper is expensive now. No wonder people steal copper wires from power station.
$400 for a new radiator core with repaired tanks fitted sounds reasonable to me. The cost was your new core, not the tank repair by the sounds of it. If he needed to replace the core, you probably did yourself a favour .... A new radiator is better than an overheated engine
With regards to a level sensor ....
The level warnings on ebay seem incredibly expensive. I was thinking something like this:
Mini Vertical Stainless Steel Liquid Water Level Sensor Horizontal Float Switch | eBay
screwed into the top radiator tank, wired into a simple hidden buzzer inside the car would be perfect. You can buy buzzer/piezos cheaply.
12V DC Relay Buzzer | eBay
It would be less than $20 all setup and should be reliable being a simple float valve and buzzer with no electronics.