curl (and most other programs) rely on a library and a certificate store for TLS.
To get it to fit in a router with only 4 MB of flash, you will most likely need to build your own firmware, especially as it appears that the builds with LuCI are too large for the device (as there are no current images that I could find).
The mbedTLS is one that was written with embedded devices in mind. It may be one of the smaller ones. I don't know much about wolfSSL, but it seems to be designed for embedded devices as well. OpenSSL and GNUTLS come from the desktop/server world and are probably larger, especially once dependencies are taken into account.
You will also need trusted certificates, such as provided by ca-certificates