Wagyu for 4000 is a stretch, even for lunch. You can have a lunch set for 4000 (or even less) but it would not be very enjoyable.

Department store restaurant floor usually have decent restaurants, often branch location of famous establishments, within that budget. For tempura and tonkatsu, 4000 would be towards the high end at restaurant floor. These places always take credit cards, have plastic model display for their dishes, often with multi-lingual memu and staff more likely to understand English.

One time a local took me to Tsunahachi main branch at Shinjuku. Supposed to be better than their department store branches. Waited in line for an hour. Can't say it's worth the wait.