[seqfan] Help with a(n) and a cumulative sum

[Éric Angelini]

Hello SeqFans,
Here is the draft of a strange sequence:
https://oeis.org/draft/A309151
It says: Lexicographically earliest sequence of different terms starting with a(1) = 1 such that a(n) doesn't share any digit with the cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n).
And in the Comments section: As this sequence needs a lot of backtracking, we don't guarantee the accuracy of the last 79 integers of the 1079-term b-file.
Indeed, the problem comes from the fact that some cumulative sums quickly block the extension of the sequence. This is the case with 10 (or any other sum ending in zero). 
But this is the case too with 301 after 1,2,3,5,4,6,7,8,9,10,11,12,14,22,20,23,24,30,13,15,16,18,28. 
So, we bumped quite often in a "bad sum" (on the average, the sequence was extended by 100 terms for every backtrack).
To make a prior list of "bad sums" is difficult (meaning impossible, I guess): 258002 is such a "bad sum" if you have previously used {1,3,4,6,7,9,39,49,69,79} else 258002 + 79 = 258081 would be ok.
So my questions are: could the sequence be infinite? Could a list of "bad sum numbers" be easely defined and used?
Post-scriptum:
I am working, together with Jean-Marc, on the variants: "Lexicographically [...] such that a(n) shares exacly k digit with the cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n)", with k = 1, 2, 3, ...
Best,
É.