[v24.3.x] dt/si: Dial timeouts way down in fast uploads mode

[vbotbuildovich]

Backport #26883 to branch v24.3.x. This issue was created as a placeholder because the original PR's commit(s) could not be automatically cherry-picked.
Here are the commands to execute:

git checkout v24.3.x
git checkout -b manual-backport-26883-v24.3.x-270
git cherry-pick -x 0243b2d32d 4049767fca 2c520d18fc

git push origin manual-backport-26883-v24.3.x-270
gh pr create \
  --title "[v24.3.x] dt/si: Dial timeouts way down in fast uploads mode" \
  --base "v24.3.x" \
  --label "kind/backport" \
  --head "manual-backport-26883-v24.3.x-270" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v24.3.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/26883 "

JIRA Link: [CORE-12944](https://redpandadata.atlassian.net/browse/CORE-12944)

[CORE-12944]: https://redpandadata.atlassian.net/browse/CORE-12944?atlOrigin=eyJpIjoiNWRkNTljNzYxNjVmNDY3MDlhMDU5Y2ZhYzA5YTRkZjUiLCJwIjoiZ2l0aHViLWNvbS1KU1cifQ