Reality is, the source button is just a programmable button. You can program it to do anything you want.
Unfortunately, your issue is that the implementation of a "source change" on your TV, requires you to press enter. This is not very common (at least not on the TVs that I have seen). If you can achieve a source change with a single button from your original TV remote, then you should be able to program that button on the ARQ-100. If you need more (like in your case, hitting enter), you would be out of luck. On most TVs, pressing the source button just cycles between the input without requiring additional confirmation from the remote. In some cases, it will show a menu which allows it to select inputs, and there is an auto timeout that if you stay on same selection for more than 2 or 3 seconds, it auto selects and switches to it.
Does you original remote have a "quick" discrete button for switching to a specific input ? I know my old sharp TV had a source button (to switch between inputs), but also had 2 other buttons which were direct shortcuts (one direct for TV, and the other for HDMI1).
Unless your TV has a single button switch option, I think you would be out of luck.
UNLESS... If you don't use the TV power button (depending on habits - some people always turn TV on using TV remote, you can always program the "TV power" button, as the TV Enter key.