Is it true that every sufficiently large integer can be written in the form $$ 2^a3^b5^c+2^d3^e5^f $$ for some integer $a,b,c,d,e,f \ge 0$?
No, this is not the case. The number of such possible sums $\le N$ for large $N$ is far less than $N$.
The number of powers of $2$ (or of $3$ or $5$) up to $N$ is $O(\log N)$. Hence the number of products $2^a 3^b 5^c$ up to $N$ is $O((\log N)^3)$. The number of sums of pairs of such products is $O((\log N)^6)$, which is $o(N)$.
No numbers congruent to $71$ or $119$ mod $120$ can be represented in this way.