First solution - perfect match but expensive.

Asus prepared solution which is perfect for your requirements but it's really expensive and it's over your budget. It's ASUS CS-B:

It meets your requirements:

Dual LAN:

• Intel® I217LM
• Realtek® 8111G

USB Ports

• 2 x USB 3.0
• 4 x USB 2.0

It's mATX.

But:

Cost: $200 (Amazon US)

Second solution - better and cheaper

You should just use @john kieran solution.

First solution - perfect match but expensive.

Asus prepared solution which is perfect for your requirements but it's really expensive and it's over your budget. It's ASUS CS-B:

It meets your requirements:

Dual LAN:

• Intel® I217LM
• Realtek® 8111G

USB Ports

• 2 x USB 3.0
• 4 x USB 2.0

It's mATX.

But:

Cost: $200 (Amazon US)

Second solution - better and cheaper

You should just use @john kieran solution.