I thought it would also be good performance-wise
Bad assumption! A 2048x2048 texture will take up a huge amount of VRAM. Construct is 100% hardware accelerated, which uses the graphics card to render the game, and VRAM (the memory on the actual graphics card) is an important consideration when creating a game. A 2048x2048 texture uses (2048 x 2048 x 4 bytes/pixel) 16mb of VRAM. Some people still have crusty old 32mb video cards hanging around, and these tend to be half full of system stuff already, so you might have already eliminated those users from being able to play your game already - with one background texture.
When designing a hardware accelerated game, it's always better to make levels using many small textures rather than single big ones. It's even better if you can do as much as possible with a Tiled Background object with a power-of-two size texture (eg. 128x128, 256x256...). You may think it would slow down rendering a bit having lots of small images on top of each other, but in many cases, surprisingly, you may actually measure zero change in the framerate! (This is because of the way the CPU and GPU work in parallel together) Even if you can find a difference, it's such a tiny change it's not worth bothering with at all: graphics cards can render simple images shockingly fast, generally only things with effects will impact the framerate.
As for your filesize issue, Construct doesn't care what the source image is when you import it. It copies it to an internal image, so the format or optimisation of the source file makes no difference.
Edit: linkman, images are compressed PNGs when they are stored in Construct, but they're lossless but not optimized.