I have done the mystery shopper call on many sign shops over the last 20 years, both franchised and independent, and rarely if ever find a franchise shop undercutting anyone. Most often they are right in the middle. One reason a franchise would be reluctant to undercut others is because the owners have to make a major outlay of capital before they can even open their doors, over $100K in 1990 when my parents opened their store. Also, the corporate headquarters get a franchise fee of about 15% (or thereabouts up to a set maximum limit) so right there is something a franchise has to add to their costs on top of all the other operating and material costs.
When I was 19 I helped my parents start a franchise sign shop and I think the biggest 'advantage' any franchise has, whether it be a sign shop, a restaurant or a retail store, is simply brand recognition. Think about what you do during the day and ask yourself "How many stores, restaurants, etc. do I go to that are not a franchise?" You do this because you are not willing to risk your money on something you are unfamiliar with. It is not until you have confidence in a store or some other reason (i.e., specials that are being run) that you will do business with it.
One point of branding is to install confidence in the consumer that what I am buying is of a certain quality without having ever experienced it. As a franchise you have a national brand and therefore are perceived as more of a professional establishment.
Case in point, when I went to California for the first time I was with a salesman and we stopped for lunch at a Jack-In-The-Box. Having never even heard of JITB I would never have stopped there had it not been for the salesman I was with, I had never heard of JITB and had no idea if the food was good or not. I would have gone to McD's, BK, Subway, etc., something I was familiar with and had some brand recognition with me.
Having gone through the setup of a franchise sign shop (not FastSigns however) I know one of FastSigns key marketing tools is print and TV advertising, where the franchise I trained under was more about cold calls and personal introductions. Both have the up/downs but when it comes to a first time buyer I would probably think of FastSigns having seen their ads on TV.
The truth of the matter however is that each sign shop is only as good as those working behind the counter. As with every business no amount of advertising or specials or anything else can override the value one derives from the quality of service or product. I am very happy with BestBuy and its service and return policy and as such am willing to pay a little more than say a Sears or HH Gregg.
All that being said (whew, that was a lot) you need to determine how to get your message out to those that are unfamiliar with you. When I started my own sign shop I worked hard at getting out door to door and introducing myself in person. I believe that I have a good personality and can make a good impression on a potential customer. This is not a high pressured sales call but rather just an introduction to give the potential client my information, a 5x8 printed intro card listing my services and a business card. It is rare that anyone needs anything when I first call on them but often they will remember someone that took the time to come to there place of business personally versus some ad they see on TV. I also get the point of contact and will call them every few months just to remind them about me and see if there is anything I can do for them.
I also offer referral bonuses to clients who refer someone to me. Most importantly (at least I think) is that I routinely contact my existing customers via personal phone call, personal visit or mass emailing. Once I get a customer to try me once I do everything I can think of to keep them.
(Please don't take the following as me saying FastSigns is better as I have no idea of the quality of work on either yours or their end.)
And after doing all that it may just be that the competitor is simply better than you. It is very difficult to honestly critique yourself when it comes to this matter. This is probably the hardest thing to quantify even from feedback as often times a customer is uncomfortable giving honest critique. Instead they may just go elsewhere and never tell you why. I once asked, via mass email, for my customers to provide a review of my services and unfortunately did not get many responses and those that did were all positive so I don't put a lot of confidence in the accuracy of the poll.