Rust around the drain in the larger of the two sinks

shut off the exit, fill with vinegar. leave overnight

If the clips are plastic you will have to snap them off. the metal ones you might have to cut them off. With the old sinks, when the screws were broken or rusted you would have to cut the clips. NO Easy job.

Replace the drain and plug with a new one while using silicone to seal the upper and lower half of the drain.........It is also a good practice to lubricate the rubber washer as well...........Wait 24 hours and it should be good to ge........Hope this helps.

Here https://www.amazon.co.uk/Franke-Kitchen-Drainer-Stainless-112-0050-236/dp/B00XVYTU12 you can get the Franke inside sink dish drainer but honestly speaking, it's good a right option and I would recommend you to go with over the sink dish drainer like this https://www.amazon.com/gp/product/B07RQMPY7Y. Over the sink, ones are very reliable especially when you do dishwashing manually. However, choices are all yours.

All that is is sound deadener to make the sink quieter with dish washing. You can use any sort of high tack glue. I am fond of household goop. Mine fell off long ago and I tossed it. never noticed a difference.

I have spent several hours trying to solve the same problem with a Franke Granite double basin sink. The clips for this Franke sink appear to be a piece of engineering garbage, and the instructions are utterly useless - A huge piece of paper with a template, and tiny pictures that look like they were drawn by a kindergartner with a crayon. The instructions for using the clips --oh there are none - other than a blurry picture in which the clips could fit inside this period (.). Lots of other people appear to have the same issue. I like the Franke sink, but without proper mounting hardware, this franke sink is no better than the poorly made clips, and the Franke Sink horrible instructions that it came with.

Take it back out and clean all of the plumbers putty off. make sure that the bottom of the sink and the basket is clean and dry. Take a good quality silicone caulk and make a 1/4" bead all the way around the basket where it meets the sink. Tighten the basket nut till it gets nice a snug but not os that you spin the basket much. Then let the caulk setup over night. Trim away the excess with a sharp knife from the top and hook up the drain. This will seal a basket everytime. I never use plumbers putty anymore.

It's what some contractor likes. If you find a sink that you like, buy it. ANY decent contractor can easily and successfully mount ANY sink.

Once that fitting is damaged it is very difficult to repair. Try contacting Franke for nearest distributer and see if they will replace.

You should purchase a complete basket strainer assembly. Home Depot or Lowes have some very nice ones. They are a standard size.
Tell them it's for your kitchen sink.