Yes I'm using FatFS to store files. But I'm also working with httpserver_raw from Texas Instruments TivaWare library and there is no LWIP_HTTPD_DYNAMIC_FILE_READ setting. Maybe this is some old example of http server. But this issue is related also to a last httpd.c from lwip v2.0.x apps. I didn't test it with static files which are stored in internal flash memory.
For a v2.0.x the changes should be made in http_send_data_ssi function in a 1209 line:
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while((ssi->tag_state == TAG_SENDING) || (ssi->parse_left) && (err == ERR_OK)) {
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I found a trouble in SSI handling in httpsever_raw application of lwip v1.4.1 and the same issue presented in v2.0.x too.
The issue is appeared if you have a long *.shtml file which size more of equal to 5840 bytes and if you have SSI tag which will has its closing '>' symbol in a 5840-th byte. In this case the hs->tag_state will switch to TAG_SENDING state and we will go to a next revolution of 'while' loop. But the hs->parse_left = 0 here and we will go out of loop without sending this last tag. And we won't send the continue of *shtml file too. It will ends here.
I've catch this situation in debugger see the attachment.
I'm also found a solution:
I've add the checking for a TAG_SENDING state to the loop expression.

    /* We have sent all the data that was already parsed so continue parsing
     the buffer contents looking for SSI tags. /
    while((hs->tag_state == TAG_SENDING) || (hs->parse_left) && (err == ERR_OK)) {

The same fix can be applied in appropriate place in lwip v2.0.x httpd example.