If you work under a Designated Activities visa, more specifically under working holiday and get officially hired by your current employer, you need to switch to an appropriate visa such as an Engineer/Specialist in Humanities/International Services visa.
You are now allowed to change your status in Japan if you are staying in Japan for working holiday. You must go outside Japan and apply for a visa at an embassy after receiving a certificate of eligibility. However, you can apply for a certificate of eligibility while in Japan.
Your current residence card is not necessary for applying for a certificate of eligibility, but inspectors do not need to check your current activities in Japan if your residence card is submitted, and that might accelerate your screening process a bit.