Not at this point.
The three 4K channels that DIRECTV has (channels 104-106) only work on 4K TVs. This is not a situation like HD where the receiver is able to downsample the picture. You've seen this before if you connect an HD receiver to an SD TV - you get an SD picture and you even have some options for how the picture looks. In the case of 4K, you'll just get a black screen with the message you see above. In most cases, if the program is also available in HD, you'll be able to watch it on another channel, but if it's only available in 4K, you'll need a 4K TV in order to watch it.
It's hard to know why DIRECTV went this route where they happily downsampled the content in the past. It could be that there's some reason hidden in the contract language, It's possible, though not likely, that DIRECTV's DVRs don't have enough "oomph" to create an SD or HD picture from a 4K stream. For whatever reason, they just don't do it. It may be that it really doesn't end up being a big deal, but for now there's no way around it.