It turned out to be quite easy to patch DOSBox to allow using a keyboard key as a mouse button. I submitted a patch for adding that functionality to the key mapper. Recently (Nov 2023) it was merged into dosbox-staging, and this change is due to be included in DOSBox Staging 0.81.

If you want to try this out, save the following into a text file called map-mouse-buttons.diff:

Index: src/gui/sdl_mapper.cpp
===================================================================
--- src/gui/sdl_mapper.cpp  (revision 3914)
+++ src/gui/sdl_mapper.cpp  (working copy)
@@ -32,6 +32,7 @@
 #include "dosbox.h"
 #include "video.h"
 #include "keyboard.h"
+#include "mouse.h"
 #include "joystick.h"
 #include "support.h"
 #include "mapper.h"
@@ -1497,6 +1498,20 @@
    KBD_KEYS key;
 };
 
+class CMouseButtonEvent : public CTriggeredEvent {
+public:
+   CMouseButtonEvent(char const * const _entry,Bit8u _button) : CTriggeredEvent(_entry) {
+       button=_button;
+   }
+   void Active(bool yesno) {
+       if (yesno)
+           Mouse_ButtonPressed(button);
+       else
+           Mouse_ButtonReleased(button);
+   }
+   Bit8u button;
+};
+   
 class CJAxisEvent : public CContinuousEvent {
 public:
    CJAxisEvent(char const * const _entry,Bitu _stick,Bitu _axis,bool _positive,CJAxisEvent * _opposite_axis) : CContinuousEvent(_entry) {
@@ -1708,6 +1723,15 @@
    return event;
 }
 
+static CMouseButtonEvent * AddMouseButtonEvent(Bitu x,Bitu y,Bitu dx,Bitu dy,char const * const title,char const * const entry,Bit8u button) {
+   char buf[64];
+   strcpy(buf,"mouse_");
+   strcat(buf,entry);
+   CMouseButtonEvent * event=new CMouseButtonEvent(buf,button);
+   new CEventButton(x,y,dx,dy,title,event);
+   return event;
+}
+
 static CJAxisEvent * AddJAxisButton(Bitu x,Bitu y,Bitu dx,Bitu dy,char const * const title,Bitu stick,Bitu axis,bool positive,CJAxisEvent * opposite_axis) {
    char buf[64];
    sprintf(buf,"jaxis_%d_%d%s",stick,axis,positive ? "+" : "-");
@@ -1866,6 +1890,15 @@
    AddKeyButtonEvent(PX(XO+2),PY(YO+4),BW,BH,".","kp_period",KBD_kpperiod);
 #undef XO
 #undef YO
+#define XO 5
+#define YO 8
+   /* Mouse Buttons */
+   new CTextButton(PX(XO+0),PY(YO-1),3*BW,20,"Mouse");
+   AddMouseButtonEvent(PX(XO+0),PY(YO),BW,BH,"L","left",0);
+   AddMouseButtonEvent(PX(XO+1),PY(YO),BW,BH,"M","middle",2);
+   AddMouseButtonEvent(PX(XO+2),PY(YO),BW,BH,"R","right",1);
+#undef XO
+#undef YO
 #define XO 10
 #define YO 8
    /* Joystick Buttons/Texts */

Then, in the terminal, check out the latest DOSBox code from Subversion:

svn co svn://svn.code.sf.net/p/dosbox/code-0/dosbox/trunk dosbox-svn

Enter the directory, and apply the patch with the patch command:

cd dosbox-svn
patch -p0 < ../map-mouse-buttons.diff

(This assumes that the patch file is saved in the parent directory of dosbox-svn. If it's somewhere else, adjust the command as needed.)

Then, follow the instructions in the INSTALL file to build your new version of DOSBox. Basically, it's these three commands:

./autogen.sh
./configure
make

Though you may have to install various dependencies, which can be done through Macports and similar tools.