What sounds like might be happening is that you have two different DHCP systems fighting. When you computers are on your ISPs modem/router and a computer sends out a IP request only the modem/router can respond. However when your going though your modem/router and then router your getting two DHCP offers depending on what is going on with the rest of the network.
That could be further complicated by the way that the ISP has setup their DHCP and how they handle when your trying, inadvertently, to grab more than 3 IPs. Suffice to say it's easy to see why it's a mess for you right now.
While it would be nice to keep the 3 IP system given what I know from your posts I think it would be easier if you were to move to a single static IP from your ISP and then just have all the computers use a NAT. My guess is that your modem/router from your ISP could handle that and you would not need the D-Link router at all.
Now if you wanted to keep the 3 real IPs then your going to have to figure out how you want to manage your DHCP setup. Off the top of my head I'd say maybe:
1. Turn off the DHCP on the D-Link router. Set it up such that users can just grab a reserved IP and assign one to each of the computers. (192.168.1.1, 192.168.1.2, etc)
2. Multi-Home all of the computers NICs with a DHCP setup and then with the static setup as well.
3. Depending on what your plugging into enable/disable the NIC. You might be able to get away with leaving both enabled but again it depends on how the ISP handles requests for additional IPs once your over the limit.
There is I'm sure a more elegant solution but there is one off the top of my head for you.

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