How many LUTs are needed to implement a CPU?
An example https://opencores.org/project/zipcpu :


  The OpenRISC processor, however, is complex and hefty at about 4,500 LUTs. It has a lot of features of modern CPUs within it that ... well, let's just say it's not the little guy on the block. The Zip CPU is designed to be lighter weight, costing only about 2,300 LUTs with no peripherals, and 3,200 LUTs with some very basic peripherals. 


Someone has build a small emulator for the 8-bit 6502 http://www.aholme.co.uk/6502/Main.htm:


  This core manages performance equivalent to a 10 MHz 6502 using less than 700 LUTs in a Xilinx xc3s500e Spartan-3E FPGA. 


That number seems very reasonable given that that 6502 only used about 3,000 transistors.

So the Lattice Crosslink should suit you with plenty of space for peripherals even for a 32-bit CPU, and is far larger than needed for an 8-bit simple CPU.