You may be able to make a board smaller in 4-layer so the square inches may compensate somewhat. 4 layer has dropped quite a bit in price in the last couple years, price starts to tick up considerably at 6 or more.
That said, 2 layer looks do-able. There was a layout here some time ago which has a number of examples of what not to do (pours are not great, power lines don't look great, crystals are very poorly and unnecessarily placed in relation to other lines). Make sure that the freelancer follows the hardware guide.
Personally, I would go for 4-layer unless I had a lot of confidence in the designer and budgeted enough of their time to take care with signal and power integrity (assuming they know how to do that in the first place). On the other hand, a novice following the guide would likely not have any issues with a reasonable 4-layer or more layout. I've seen some pretty slapdash subcontractors purveying their work online.