If you haven't already scoped out your local security (anti-virus, firewall, router's security/WAN settings, etc.) to make sure nothing there is blocking you, then it sounds like an issue with the routing your ISP is assigning you to get to SE's servers in Montreal and/or Japan (you authenticate against servers in Japan briefly). You will need to push through to your ISP's Tier3 support team and have them run some tests for you (or you can run some traces yourself and forward them via email or chat). They will need some addresses that will point them to SE's servers so they can get a clearer picture of what is going on. Here are the registered DNS names for the current lobby servers (note, these may change when they implement the servers in Europe):

Lobby servers in Canada for NA/EU:
neolobby02.ffxiv.com
neolobby04.ffxiv.com
neolobby06.ffxiv.com

Lobby servers in Japan:
neolobby01.ffxiv.com
neolobby03.ffxiv.com
neolobby05.ffxiv.com

In the short term, you could try running with a VPN service to see if things improve. Many have free trials or even free accounts that you can use on a regular basis (will have caps on how long or how much data you can use per day/week/month on the free accounts). It is worth it to at least try the free account or free trial periods for them because it is a way to show your ISP that it is indeed an issue along your route. Using a VPN basically sidesteps the policies negatively impacting your route. Basically, you can show them that things are fine while you are running an alternate route or otherwise avoiding their shaping rules but still on the same modem and wire, demonstrating that there is something wrong along the route they have assigned you to get to the same servers. That is something they are responsible to investigate and take steps to correct.

A lot of people here use the gaming oriented services like WTFast, Battle Ping, and Ping Zapper, but there are other "generic" services you can try. Here are some notes I took on a handful of services I tested back during Beta4 when we were having a lot of trouble with routing early on.

https://www.tunnelbear.com
500MB/month free. Has Canadian server.
$5/month or $50/yr for unlimited bandwidth.
****FFXIV:ARR WORKS, BUT CA CONX WAS DROPPING, US SEEMS MORE STABLE?***
****SPEEDTESTS WERE STRONG, 14/1, GOOD PINGS***

https://cyberghostvpn.com/en/product/purchase.html
No Canadian servers, New York and Washington are listed with their free service.
Free plan has a 3 hour connection limit, uses OpenVPN client.
Premium $7/mo, $40/yr, $70/2yrs (one device--Plus gives 5 for $11/70/110)
***Free plan had an ad when connecting, another popped up after @ 2 hours***
***NY connection timed out in tracerts, WA seems to respond well***
***FFXIV:ARR CONNECTS, WILL NEED TO TEST QUALITY, AS PINGS NOT AS GOOD AS TUNNELBEAR****
***SPEEDTEST SHOWED 6/3 SPEEDS, LOW PINGS TO MONTREAL--SHOWS AS A UK CONX??!!***

http://www.hotspotshield.com/
has free VPN, and free proxy--both ad supported.
Ads may pose issues with games if screen looses focus due to a pop-up?
Elite: $30/yr or $20/6 months.
***no Canadian server, and Free is US only.
***FFXIV:ARR CONNECTS WHILE IN 3-DAY ELITE TRIAL, BUT IS THROUGH LA AREA. MAY HAVE ISSUES?***

https://www.sumrando.com/why.aspx
free is 1GB cap, server in New York. (did not note if the 1GB renews monthly, or is just a one time thing)
***FFXIV:ARR CONNECTS, BUT LOOKS TO STILL BE HITTING SAME LEVEL3 HOPS HAVING ISSUES***
***SHOWS UP AS WICHITA, KS AT SPEEDTEST--BENCHED 9-12/1 AT EXIT POINT, UNEVEN GRAPHS***
***SAW SOME LAG SPIKES--TRACERT SHOWED SAME TROUBLE AT CAR2 ON LEVEL3 HOP***

Back in Beta4 and early launch, I started using Tunnel Bear to prove the point to my ISP. It wasn't the greatest, but they had an exit point in Montreal--and it was a real eye opener for them. Should note that the stability issues to the Montreal exit point have been resolved--I've used it for extended periods of time doing business and it no longer drops like it was doing last fall when I made those notes. Their free account gives you 500MB/month (1GB if you promote them on Twitter), can get it for under $5 a month on their annual plan, sometimes less if they run a special. I have kept the free account to use for some things I want to do on a secure line, so it may be something to look into if you feel you have a need for a VPN from time to time like I do.