No. Article I, Section 3: "Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States." The Democrats wrote the impeachment indictment to prevent Trump from ever holding office again if convicted.
But he's not going to removed so it's a moot point.
You are reading the clause wrong. The senate, if they convict, “MAY” remove from office and “MAY” disqualify him from holding any office. A prime example is Rep. Hastings, (a former judge), who was convicted and removed but not excluded from holding “office”.
The house has no say in what punishment the senate can impose since the senate has sole authority to hold the trial and determine the punishment, (just as a prosecutor can’t force a judge to sentence the defendant to a specific punishment after a guilty finding, neither can the house).