Yes, airlines can use a 156 seat layout in the A319. See Allegiant's layout.
The FAA type certification for the A320 series can be found on the site here, under TCDS A28NM. For the A319:
See note 11 and 18
If modification 32208 is embodied in production on A319 airplanes, the aircraft is eligible for maximum capacity of 160 passengers. This modification consists in structural and system provision for the installation of second pair of overwing emergency exits.
The certification basis section g states that the design complies with the following regulations:
- 25.811(e)(3) Type III emergency exit marking.
- 25.813(c) for emergency exit access
The regulation from section 25.813:
§25.813 Emergency exit access.
(c) The following must be provided for each Type III or Type IV exit—(1) There must be access from the nearest aisle to each exit. In addition, for each Type III exit in an airplane that has a passenger seating configuration of 60 or more—
(i) Except as provided in paragraph (c)(1)(ii), the access must be provided by an unobstructed passageway that is at least 10 inches in width for interior arrangements in which the adjacent seat rows on the exit side of the aisle contain no more than two seats, or 20 inches in width for interior arrangements in which those rows contain three seats.
The A320 series has three seats on each side, so the applicable distance is a 20 inch wide passage. The referenced paragraph (c)(1)(ii) is also interesting:
(ii) In lieu of one 10- or 20-inch passageway, there may be two passageways, between seat rows only, that must be at least 6 inches in width and lead to an unobstructed space adjacent to each exit.
This appears to be one layout that Air Canada uses.
There doesn't appear to be any regulation requiring a certain number of lavatories. Aircraft sometimes fly with only one working. Evidently no US agency requires even at least one.
However, the DOT does appear to have a rule created by the Air Carrier Access Act that says:
New twin-aisle aircraft must have accessible lavatories.
I read that as requiring the aircraft to have at least one, possibly two, lavatories, and that they must be accessible.
"New aircraft" requirements apply to U.S. airlines with respect to planes ordered after April 5, 1990 or delivered after April 5, 1992. In general they apply to foreign carriers with respect to aircraft ordered after May 13, 2009 or delivered after May 13, 2010.
Of course this wouldn't apply to the A320 series, as it is a single-aisle design. Although some twin-aisle aircraft do operate on shorter routes, they are probably less likely to go the route of ">200 people, no toilets."