How safe is a prime with $P=2 \cdot Q \cdot R \cdot S \cdot t+1$ for discrete logarithm? How to enhance/compare?
It does work with Silver-Pohlig-Hellman algorithm

As theREALyumdub pointed out in comments that Silver-Pohlig-Hellman might be an option I did some test and it did work. Thanks for that hint. 

In case $t=1$ we get the results $a,b,c$ right out of the algorithm. For other $t$ the results $a,b,c$ aren't correct. Some extra work need to be done there. It does work if you multiply value $v$ with the inverse of used $T$ first. However this $T$ is unknown in use case. Chances are high there is a better way but the naive way is good enough for my case. As I wrote in post $t&lt;1000$ so we could just multiply with a random number and get the correct result with chance $\frac{1}{t}$.

So it is not harder than:
finding d for a given $v'$

$v'\equiv g^d \bmod P$, with $g$ generator of a subgroup with size $G=\max(Q,R,S)$

Remaining questions:

Does it make a difference if you solving only for a subgroup? Or is equation above, as hard as
$v'\equiv g^d \bmod G$

Is there any way to speed up finding the correct $T$ factor? While a 
constant factor of 1000 is asymptotically equal but in real world it can make a huge difference.

However this transition used in Silver-Pohlig-Hellman algorithm destroys the idea of hiding the dimensions in one number. Three separate equations with big prime factors would do the same job.

Although the question is not fully answered it's enough for me. The remaining questions above are not specific to this case. I might create some new threads about those. If so, I will link them here.

$v = T\cdot g^a\mod P$
proof if same factor $T$ was used -->  $v_{T_1}^{QRS} \equiv v_{T_2}^{QRS} \mod P$  iff same group 

links show easy to do -> this kind of prime and usage has not benefits :(