Experimenting with the approach described by FuzzyChef, will probably be faster and more convenient than the approach suggested in this answer; this answer is mostly a record of my experiments.
Caramelized White Chocolate Ganache
Many if not most white chocolate varieties use deodorized cocoa butter, so most of the flavors present are from milk solids and vanilla, with relatively little "chocolatey" flavor. Caramelized white chocolate is heat processed in a way that browns the milk solids and sugars; the end result is chocolate that tastes similar to milk caramels like dulce de leche.
Caramelized white chocolate behaves like other chocolates; as far as I can tell, it produces a ganache that is relatively close to one produced with dark chocolate.
Executive Summary of Recipe and Results
Melt 4 parts of caramelized white chocolate per one part butter (by weight) over gentle heat and stir until combined. 2 tablespoons is about one ounce of butter, so the ratio is 2 ounces of chocolate per tablespoon of butter.
Caramelized white chocolate ganache is noticeably grittier than a ganache made with the same amount of dark chocolate and butter. Out of the refrigerator, it is slightly harder than dark chocolate ganache, but, after enough time spent at room temperature, both soften to similar textures.
Caramelized white chocolate ganache will taste like the chocolate used. Especially when deeply browned, caramelized white chocolate can taste like milk caramels (for example, dulce de leche).
The process of caramelizing white chocolate at home does not appear to introduce or intensify any dark chocolate flavors, though it also does not completely remove vanilla flavors if they were present in the white chocolate before caramelization.
Between the caramel flavors and vanilla, ganache made with caramelized white chocolate will resist further flavoring.
Logistical Difficulty: Acquiring Caramelized White Chocolate
Unfortunately, obtaining caramelized white chocolate may be inconvenient. It is available for purchase from some vendors, but the most readily available option is probably making it yourself from white chocolate.
Buying Caramelized White Chocolate
Caramelized white chocolate is a specialty product; it will likely be only available at specialty stores or online, and at a markup compared to similar white chocolates. Two examples of caramelized white chocolates available online are:
Caramelizing White Chocolate at Home
It is possible to make caramelized white chocolate at home.
If you have an oven, you can caramelize white chocolate by cooking it in a 250F (120C) oven four about an hour, stirring every 10 minutes. There are more details in this recipe by David Lebovitz.
It is also possible to caramelize small amounts of white chocolate in a microwave by microwaving in 15-30 second bursts, stirring thoroughly between bursts. This method may take as much as 45 minutes for 4 ounces of chocolate and requires close attention; I find it tedious and recommend the oven method. Note that the chocolate will still reach temperatures around 250F before it starts appreciably browning; use a sufficiently heat-proof bowl.
Choosing the Right White Chocolate
Since you want to avoid the flavors of dark chocolate, you should avoid white chocolated made with non-deodorized cocoa butter. Deodorized is usually the default; El Rey is the only producer that I know of that sells bulk white chocolate made with non-deodorized cocoa butter.
In the US, Ghirardelli and Guittard are two brands that make relatively common white chocolate (*and non-chocolate "white chips") bars and wafers for baking. The varieties that I know of are produced with deodorized cocoa butter.
The recipe by Lebotitz also links to an article that makes the distinction between "real white chocolate" and similar candies that use fats other than milkfat and cocoa butter (cacao fat). This distinction is also maintained by some regulatory bodies, including the United States FDA.
The term "white chips" is used for some of these confections, which are sold alongside other chocolate chips. The presence of fats other than cocoa butter and milkfat means that these chips may have textures and melting properties slightly different from those of white chocolate. On the other hand, the smaller amount (or absence) of cacao product may be desirable if you want to minimize the possibility of introducing chocolate flavors.
In this application, caramelized "white chips" may be close enough to the real thing. Ganache produced with 4:1 caramelized white chips:butter seems to have a similar texture; I find it slightly waxy compared to a ganache prepared with white chocolate, and, when both are brought to room temperature, the ganache made with white chips seems slightly harder than a ganache produced with white chocolate.
However, compared to white chocolate, "white chips" seem more prone to burning or producing large, crunchy pieces when caramelized in a microwave. This can be somewhat mitigated by caramelizing in larger batches (at least 2 ounces or 56g at a time). I was not able to test caramelizing white chips in an oven; the gentler heat of the oven may make this tendency a non-issue.