APM site slowness/timeout

So would you be lead to believe it is a problem with the LTM VIP?

I guess that depends on where the external DMZ address is in context to the LTM VIP. Let's say, for example, you have a network path like the following:

client - Internet - router - firewall - DMZ - firewall - router - LTM VIP                

This likely isn't a correct representation of your environment, but illustrates the path that a packet might take to go from the client to the LTM VIP; and like an electronic circuit, you can place a multimeter probe anywhere along this path and "see" the electrons flowing. So then the question becomes, if you stick a probe at the LTM VIP interface when it's failing, do you see the client traffic reaching the interface? If not, do you see it reaching (and leaving) the device preceding it (and so on)? Only if you see traffic at the LTM VIP when it's failing can you assume that the problem is at/in the VIP.