For this purpose, I'd imagine you would not likely see much difference. However, there is literally no reason to prefer FPKM over TPM. If you're looking to perform some analysis where relative abundance is an appropriate measure, you should always favor TPM.
Thank you everyone for your help! I tried it both ways. The PCA from the FPKM values made the most sense and the plot was similar to previous work. When I used the TPM values the strong separation by PC1 that we had seen with FPKM and in previous analysis moved to PC2.