Protecting Ed448 against DPA and fault attacks
The, common, assumption used in that paper is that an attacker can DPA a fixed secret when it's mixed with known (to the attacker) and varying data.

The proposed solution was to add random after the key to fill the SHA-512 block size.

By doing this:


the first SHA-512 computation processes the fixed secret and varying but unknown data, so the attacker can't perform DPA. 
After the first block, the result is the intermediate hash value which is unknown and varying (from message to message), so it can't be recovered through DPA when processing subsequent blocks.


Side note: if we would have padded the key with zeros instead of random, we would have mixed the fixed key with known but fixed data. The attacker in this case is not able to DPA the key. However she can DPA the intermediate hash value after the first block, which is secret and now fixed and will get mixed with the known and varying message.

Your proposed solution SHAKE256(dom4(prehash, context) || b || Z || PH(M), 114) with Z random data would work if you carefully select the size of Z and assuming the attacker can't control the context.

If the attacker control the context then she can choose varying context that together will the key (b) will completely fill the SHAKE256 block, so no random is used in that computation and DPA can be performed.

If the attacker can't control the context then you just have to make sure that Z is not too small and fills the block.

For example, given the block size of SHAKE256 of 136 bytes, if len(dom4(prehash, context) || b)=134, due to a large context, then 2 bytes for Z won't be enough and you will need to select a Z of 138 bytes (2 to fill the current block and 136 for another full block).



Update:
Please be aware that the generation of r is one way to side-channel attack EdDSA, but it's not the only one. The scalar multiplication should be protected as well from single trace attacks (technically not DPA) and the computation of $S = (r + k * s)$ could be targeted by DPA as $s$ is fixed and unknown and $k$ is varying and known.