Note that this answer is wrong, as pointed out by amon in the comments. It is mostly correct for LGPL2, but LGPL3 added a special header exemption in section 3 that allows redistribution of an application at any terms even if there are header files with significant logic in them.
I need to rewrite or delete the remaining answer. The rewritten answer will ask the question: "You don't get the freedom for the end users tor swap out your library, so why use LGPLv3 at all?" instead of "Do you want the consumer of library to require publishing their source code that interfaces with the library?".
You, the library author, can publish your library under any valid license, and as far as we know, the LPGL 3 is considered a valid license. The question you need to focus on is not "is the LGPL OK to use for <library of type x>", but instead "does the LGPL have the intended effect on <library of type x>".
The key point of the LGPL is (in my opinion), that the users of your library can use the library in non-(L)GPL products, so your library can get used in more places. At the same time, the LGPL ensures that the users of the product where your library is incorporated still have the the same freedom for working on the code of your library and using a modified version of your library as if this third-party product were GPL licensed. This means that authors of 3rd party products needs to make sure that users of these product can incorporate a modified version of the library. With C++ header-only libraries, the only way to switch out your library with a modified version of your library is recompiling the 3rd party application. In consequence, you force 3rd party authors to publish their source code if they want to legally use your library. You do not force 3rd party authors to publish their source code under an open source license, though.
So, the question you need to really think about is: Do you intend the effect that users of your library need to publish their source code, at least of the source files that interact with your library?