I am thinking if I could setup a miner client to mine ONLY very low-valued blocks, because, I hope, there is less people trying to solve them.
Every miner is always trying to solve a different block.
If two miners are not coordinated, they are each trying to mine blocks that reward them and not the other miner. So miners who aren't coordinating are trying to solve different blocks already.
If two miners are coordinated, they will coordinate to ensure that they are not trying to solve the same block as repeated effort is wasteful. So miners who are coordinating are trying to solve different blocks already.
So trying to mine lower-valued transactions conveys no benefit. You are also trying to solve a block that only you are working on, but you get less in transaction fees if you succeed. So it's a net loss.
Even prioritizing your own transactions doesn't do you much good. Every one of your own transactions you prioritize over a transaction that would have paid you more costs you the same amount as you saved on the transaction fee.
For example, say you have a transaction that you're willing to pay a transaction fee of 100 for to ensure it doesn't take too terribly long to get in even if you don't successfully mine a block. Then say you include your own transaction instead of a transaction from someone else that had a 200 unit fee. You save the 100 units you would have had to pay to let your transaction get in eventually but you also cost yourself 200 units because you didn't get the 200 unit fee from the other transaction. Your net loss is 100 units -- exactly what you could have added to the transaction fee to get the transaction in the block had someone else mined it.
The net effect is nothing at all when you mine your own transactions. Except that you have to wait longer to get your transaction in if you don't get lucky and happen to mine a block while it's pending. So it's a net loss.