Is it possible to mirror 2.5K or 5K iMac display to 1080p TV through Apple TV without swithcing iMac's resolution?

I have a 27" iMac with 2.5K built-in display resolution. Also i may replace it with a 5K/retina one. The problem is mirroring it's display without changing my mac's resolution other than it's native resolution. When I use a Thunderbolt to HDMI converter and an HDMI cable, my mac forces me to use lowest resolution between itself and my tv (1920x1080/2K). So it prefers the tv's resolution and my mac switches it's built-in display resolution to 2K and this is not what I want. So, I think about Apple TV and AirPlay mirroring. My question is: **Is it possible to mirror my mac's high resolution built-in display to a low resolution (2K) tv through Apple TV without changing the mac's resolution?** If it's possible, naturally higher resolution image will be scaled down/transcoded to a lower resolution image. Texts and details on original image will be corrupted/blurred but it's ok, I don't expect/need a super clear image on the tv. So, will Apple TV do the job, can it transcode original high resolution image to fullscreen 2K image for my tv?