Generally speaking meeting the EMC/EMI requirements is the EE job. The biggest part of radiated EM waves usually comes from a poor PCB, and of course there is little the ME can do about that. Providing more space, if possible, can relax some constraints for the EE that will have more space to properly route his/hers tracks.
In your case it appears to me that no EM problem should be present: the fastest signal lying around would pass through the USB connection, but you say it's used only for power. Your device is also battery power and that gives a huge help since there's no risk to inject some unwanted frequencies in the mains through the power supply.
If there's some high speed clock inside, and I guess there is, proper routing should be enough. A great help would come from a metallic enclosure, as you probably know a grounded metallic enclouser greatly helps to keep unwanted EM from escaping your device. Bonus points: it also keeps unwanted EM from messing with your circuit, that might be a bigger issue.
To help with the routing, relaxing the position of the usb connector and/or the lcd can help, but really in this case it seems quite trivial to me to make a circuit that has no problems.
Without additional informations on the specific device in question I'd say just relax, make a prototype and test it, you're probably getting away with it without any problem.
