The Welcome Message in Magento 1.7 is a problem for Lesti_Fpc. You can find the reason in app/design/frontend/base/default/template/page/html/header.phtml.
<!-- ... --> <p class="welcome-msg"><?php echo $this->getWelcome() ?> <?php echo $this->getAdditionalHtml() ?></p> <!-- ... -->
The welcome message is just a function of the header block. Lesti_Fpc needs the welcome message in a seperated block. In Magento 1.8 this is solved and the name of the block is welcome. In Magento 1.7 there is a semi-solution in core, we just have to give a little help.
I have released Lesti_Fpc 1.1.5. What is new in this version? I have add a new field in the configurations. Miss Uri Params, not a really creative and good name. The idea behind this parameters is: No request with these parameters will be cached. For example:
It's getting a little bit complicated. All Miss Uri Param will be seperated by comma and every parameter is sperated with an = from a regular expression. Every request with the parameter limit as an integer will miss the Fpc. I know it makes the configurations complexer, but I guess it will help somebody.