Windows live writer wont connect to blogger

I have not had any success in resolving this issue on my laptop.  But lo and behold, the system works fine (normally) on my desktop.

This is very frustrating.  I have been pulling my hair out trying to figure out what the differences could be in configuration.

Both machines run Windows 7 Home Premium; both use Firefox 39.0 as browsers.  Both use the same Norton Internet Security software.

Have uninstalled and reinstalled Windows Live Writer several times, and toggled various settings on google/blogger security settings.

Still get 'method not allowed' whenever I hit publish.

Maddening.

Go to https://myaccount.google.com/privacy and turn on the option to connect to non secure applications.

OK; I appreciate the guidance and offers of help.  My problem continues unabated.

I am going to do the best to summarize my situation overall.

I have been using Blogger to publish my blog for 6 years or so; the blog address is this:  http://othersideofbrunswick.

I have published in the neighborhood of 1,000 posts, and have been using Windows Live Writer (WLW) virtually since the beginning.

I have had no serious 'operating' problem with this setup over the entire period.

I have two computers: a laptop with Windows 7 Home Premium, running Firefox as the browser.  This is MY computer for daily use.  The other is a desktop with Windows 7 Home Premium, also using Firefox as the browser.  This is the computer my wife uses for trivial things like playing solitaire, etc.

I live in Brunswick, Maine, and Comcast is my 'broadband" ISP; both computers are on the same Wi-Fi network; nothing has changed on this setup in years.

Sometime in late May, as I recall, I began to have problems publishing from WLW, as numerous others did and reported.  That history is clear.

At this moment, I have the publishing problem on my laptop, but not on the desktop.  I can draft the post on WLW on my laptop, and take the resulting file to the desktop, and publish it successfully to Blogger.

I have tried every suggestion seen on these and related pages; updating WLW versions, toggling the security settings on Blogger/Google, etc.   As I understand things, I have only one Blogger account, and each computer, when I click on "Publish," is making use of the same logon credentials.

To the best of my ability, each computer has the same updates installed for Windows 7, and is running the same version of Firefox.  Each uses Norton Security, which is updated to the same configuration.  I have done my best to try to discern what is different about the laptop that causes the problem, while the desktop does not have the problem.

I run the same version of CCleaner on each to clean things up.  I run the same version of Spybot on each.  Both are the free versions.

I have set up a second blog on blogger, with this address:  http://claptrapandpoppycock.

It has the same publishing problem from WLW on my laptop.

Several days ago, I went into town and signed on to a public Wi-Fi network at a donut shop with my laptop, and got exactly the same results.  No-Go from WLW.

I do not know what else I might try.  I have done all the 'restore' tricks, along with install/uninstall, and have gotten nowhere.

That's it, I think.

There is a gremlin in the mix somewhere, and I sure would like to find out what it is.

Surely someone 'expert' can help me resolve this roadblock.

Shall I assume this forum topic has 'died on the vine?'