Hi Karol,
Thank you for contacting Syncfusion support,
We can reproduce the issue at our end. Setting tabindex to -1 will prevent the focus of input box because the DropDownList popup will not be rendered inside modal dialog and this will be appended to body of DOM. If possible, you can avoid the use of tabindex property. Otherwise, we suggest you to append the popup to the modal dialog content while using tabindex property for dialog or you can override the focus method of bootstrap. Please refer the given sample
If you face any difficulties in overriding the bootstrap method please get back with more details, we will analyze and provide a better solution
We would also like to let you know that ejDialog can be used in this case as shown in the below sample:
Regards,
Keerthana