Getting Temporary Visitor -> Engineer Visa Change from Inside the country?

Hi all!

I have a Mexican passport and came to Japan a week ago to visit my bf. After a long job hunting process i got an offer! while still in Japan!

The company will sponsor my visa (engineer), so my question is. Can I wait for the CoE and make the Visa request while still in the country?

I found information saying it's necessary to exit the country and get the visa from outside, to make the change from visitor to residence card and visa holder. If that's the case could I go to Korea for a week of so and request the visa there?

I see some embassies will deny foreign nationals without a residence from that country, does anyone have experience with that?

I have a place here and would prefer doing the process inside the country and am worried that nearby countries will reject my visa request since i do not live there.

Any help is appreciated! Thanks!