Firstly, there is no way to fit an electric start without re-engineering the motor. Excess fuel out the overflow is not a good sign. It may be that the float level is too high, have you changed it by bending the tab? Generally speaking, the brass tab on the float that pushes the needle valve shut should be flat in line with the rest of the brass float holder (does that make sense?)The only way to make sure it to measure the distance away from the carb body when the float is closeing the needle valve to the centre of the floats. I don't have the specified distance here with me, but I can get that from manual which is at home if you need it. I think this is best done by someone who has some experience with carbs. If the float is lifting too high before shutting off the fuel flow, there will be too much fule in the carb, causing it to run rich. This will usually mean it will start easier from cold, which is not what you are experiencing, however.
After you get it to start with easy start (we have a brand here in NZ from Australia called "Start You Bastard"!), does it start again ok? I.e. will it start when it is warm? If it does start OK once warmed up, then it would seem that at least the choke enriching circuit is still blocked. I have found by far the best carb cleaner to be the one you buy from a Yamaha dealer with their own brand. I learnt this from a friend, Craig Faith, who has a Yamaha dealership that specialises in restoring old Yamahas for collectors as well as selling new ones. He does dozens of old Yamahas every year, some of which have been sitting for 20 years or more. If you can get some of this stuff, DON'T follow the instructions on the label which say to add it to the fuel and run the motor. The best way is to dilute the cleaner 1:1 with water, put it in a can or old saucepan, immerse the dismantled carb and simmer (not boil) for 15-20 minutes. Rinse the carb parts with water well and blow out with compressed air if you can, otherwise wash with methylated spirits (wood alcohol). Don't throw away the fluid after you have finished. It can be reused many many times. Mine has just been used for at least the tenth time this weekend to clean up the inside and outside of a set of carbs off a 1972 Honda 500/4 that had not run for 6 years. They look like new inside and out and worked perfectly afterwards. The cleaner does not touch any rubber parts.

As far as parts go, I think your dealer can't be bothered looking as the TS185ER is still a current model. I bought mine new recently. If you want, I can source new parts and can probably get you a good second hand carb, set it up on my bike to make sure it is OK, and send it to you. Let me know about the starting after the bike is warm, and I may be able to help with any further diagnosis of the problem. Hope this helps.
Kind regards.

Nigel in NZ

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--"How can I be lost if I don't care where I am?"--

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The mouth of a perfectly contented man is filled with . -- 2200 BC Egyptian inscription