ST tires as I understand it are supposed to be inflated to the pressure listed on the tire, not something else. Otherwise you just have excessive sidewall heating which increases rolling resistance and tire wear.

Many folks feel that load range E tires are too stiff for lighter trailers like rPods (resulting in excess vibration of the trailer) and prefer to use LRD's. LRD's are typically inflated to 65 psi vs E's to 80. So 45 psi on E's is way lower than they'er designed for. 

If your tire shop thinks that your trailer is too light to inflate LRE's more than 45 lbs when they are designed for 80 psi then why did they sell you E's to begin with? Doesn't make sense to me other than that they get a higher price for the E's. 

FWIW I run LRD's on my 179 and inflate them to 65 as specified on the tires. 

Load range and tire inflation is noted on the side of your trailer.  If you have LR-D tires, the sticker will tell you to inflate the tires to the maximum stated on the tire, e.i. 65 psi.  It affects the weight capacity and the speed at which you can tow.  Trailer tires are made to run at the pressure stated on the sidewall.  To inflate them less, means that you are causing flexion in what is supposed to be very stiff tire.  Flexion causes heat and heat is the enemy of tires.  Personally, if I had LR-E tires, I'd inflate them to pretty close to or at the pressure stated on the sidewall.  That's their design inflation.

If in doubt, call a trailer tire manufacturer and get connected with one of their tire designers and ask.  You're much more likely to get accurate information than what the local tire ship guy can give.

Your tire shop is right, just like cars and trucks, tires on trailers should be inflated to match the weight. Your 178 is around 3300 lbs that's only 1650 lbs per tire and your tires have a load index of 117 = 2833 lbs or more and your rims are only rated to 1825 lbs not sure why people go to higher load index. My Toyota Tacoma takes 29 lbs in the tires (tire max is 65 lbs) and I bring it up to 36 lbs in the rear when towing. Your 178 states 50 lbs so I would think 45 lbs in a high rated tire is right. 

I use the chalk method on all my tires, too much pressure is also bad for the tire and trailer.

http://www.intercotire.com/using_chalk_method_determining_psi - http://www.intercotire.com/using_chalk_method_determining_psi

LRC are going to be around 50 PSI, LRD are about 65 PSI, and LRE is 80 PSI. I believe that there "may" be a consideration for a LRE tire operating with a very light load, but I'm not so certain that the pressure would be as low as 45 PSI.

Two seemingly opposite approaches. Which is right? Answer: both, it depends on the manufacturer. Read this:

https://www.etrailer.com/question-266105.html - https://www.etrailer.com/question-266105.html

So, go with what your specific tire manufacturer recommends. But for the 178 the axle rated load is 3500 lbs so I'd assume 1750 lbs per wheel unless I'd actually weighed it. Note that a higher inflation pressure also reduces rolling resistance which improves fuel economy. 

If you are in doubt about tire pressure, weigh your typical trailer load when in actually in use and you can see what your inflation pressure should be from the manufacturers' charts. 

e-Trailer advisors tend to opt for inflating to the maximum pressure, but if you are traveling with an especially light load, then it is probably okay to run a little lower pressure.  Ultimately, you want the tread of the tire to sit as flat as possible on the road surface and minimize sidewall flexing.  If you are a little over inflated, you will put a slightly less tread contact on the sides as compared to the center of the tread surface, but you will also minimize sidewall flexion and overheating.  If you run slightly under inflated, you will put much more stress on the sidewalls and will induce some tread squirm that causes more tread wear and heat build up.  So it is probably better to err on the side of slight over inflation, but not over the maximum stated on the tire.