I noticed that KM doesn't work properly with Google profiles. Google even provides an error message saying so, suggesting that I get a different browser. It also fails with Google Picasa, probably for the same reasons.
That would be helpful to fix this, so that I can just use one browser (KM).

Which version of KM are you using? Some pages seem to require a later display engine than the one in 1.5.4. KM 1.6.0 beta2 is very stable. I have that one as my default.

Alas I'm using the latest (1.6.0).

I went to both sites and did not see any message about a different browser. Since I do not have an ID at either, I could only go to the main page. Is it some other page where you get the error?

Here is a page from Google:
http://www.google.com/support/profiles/bin/answer.py?hl=en&answer=1151309&p=browser_support

But you have already tried the usual fixes, like switching user agent to Firefox, did you...?
Cannot test, your link is dead, found another similar one, but see no messages about "wrong browser" or such...

Thanks for looking into this. I put the following string in custom user agent:

Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.9.1.18) Gecko K-Meleon/1.6.0 not Firefox/3.5

Do you agree with this? Maybe it can be simplified? I just cut/pasted it from a message in this forum.

Also, to get the error message, just go to profiles.google.com
I'm logged into my profile, and it says it doesn't support the browser, and then I clicked on "Learn More".

There's no need for me or anyone to agree, just play a bit
Sure you can try that string, or also try the default fake for FF UA, the first entry in the UA-menu:
Tools > User Agent > Firefox 3.5

I see that google support page says they accect FF 3.6 that would be equivalent to KM1.7 (alpha), not KM1.6, or do I confuse that again??
Anyway, if the other strings don't work, try a FF3.6 string, tons can be found here as examples:
http://www.useragentstring.com/pages/Firefox/
Just a random pick:
Mozilla/5.0 (Windows; U; Windows NT 6.1; en-US; rv:1.9.2.3) Gecko/20100401 Firefox/3.6.3

If that still doesn't work, KM1.7alpha could be tested, but I don't suppose it's that complicated

Older strings aren't going to work. Google is dissing us badly. I finally got into all my G-stuff using this string:

Mozilla/5.0 (Windows NT 5.2; rv:2.0.1) Gecko/20100101 Firefox/4.0.1

A.K.A. Arual the Wyrd